Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 10.6s

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: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}\right)}^{-1}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.998:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{t\_2}}{\sin ky}}\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;t\_3 \leq 0.9992:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (pow
          (/
           (*
            (/ (hypot (sin kx) (sin ky)) (sin ky))
            (fma (* th th) 0.16666666666666666 1.0))
           th)
          -1.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.998)
     (/ (sin th) (/ (sqrt t_2) (sin ky)))
     (if (<= t_3 -0.1)
       t_1
       (if (<= t_3 0.005)
         (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
         (if (<= t_3 0.9992)
           t_1
           (*
            (/ (sin ky) (fma (* (/ 0.5 (sin ky)) kx) kx (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow((((hypot(sin(kx), sin(ky)) / sin(ky)) * fma((th * th), 0.16666666666666666, 1.0)) / th), -1.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.998) {
		tmp = sin(th) / (sqrt(t_2) / sin(ky));
	} else if (t_3 <= -0.1) {
		tmp = t_1;
	} else if (t_3 <= 0.005) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else if (t_3 <= 0.9992) {
		tmp = t_1;
	} else {
		tmp = (sin(ky) / fma(((0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(hypot(sin(kx), sin(ky)) / sin(ky)) * fma(Float64(th * th), 0.16666666666666666, 1.0)) / th) ^ -1.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.998)
		tmp = Float64(sin(th) / Float64(sqrt(t_2) / sin(ky)));
	elseif (t_3 <= -0.1)
		tmp = t_1;
	elseif (t_3 <= 0.005)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * sin(th)));
	elseif (t_3 <= 0.9992)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[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], -1.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.998], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$1, If[LessEqual[t$95$3, 0.005], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9992], t$95$1, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -0.998

   1. Initial program 94.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 28.1

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 28.1

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.1

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

   10. Applied rewrites 93.1%

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

   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))))) < -0.10000000000000001 or 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999199999999999977

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

   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 55.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.1%

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.999199999999999977 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;{\left(\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9992:\\ \;\;\;\;{\left(\frac{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky} \cdot \mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.998:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{t\_2}}{\sin ky}}\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;t\_3 \leq 0.9992:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)) -1.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.998)
     (/ (sin th) (/ (sqrt t_2) (sin ky)))
     (if (<= t_3 -0.1)
       t_1
       (if (<= t_3 0.005)
         (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
         (if (<= t_3 0.9992)
           t_1
           (*
            (/ (sin ky) (fma (* (/ 0.5 (sin ky)) kx) kx (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow((hypot(sin(kx), sin(ky)) / (sin(ky) * th)), -1.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.998) {
		tmp = sin(th) / (sqrt(t_2) / sin(ky));
	} else if (t_3 <= -0.1) {
		tmp = t_1;
	} else if (t_3 <= 0.005) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else if (t_3 <= 0.9992) {
		tmp = t_1;
	} else {
		tmp = (sin(ky) / fma(((0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)) ^ -1.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.998)
		tmp = Float64(sin(th) / Float64(sqrt(t_2) / sin(ky)));
	elseif (t_3 <= -0.1)
		tmp = t_1;
	elseif (t_3 <= 0.005)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * sin(th)));
	elseif (t_3 <= 0.9992)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(0.5 / sin(ky)) * kx), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.998], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$1, If[LessEqual[t$95$3, 0.005], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9992], t$95$1, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -0.998

   1. Initial program 94.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 28.1

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 28.1

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.1

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

   10. Applied rewrites 93.1%

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

   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))))) < -0.10000000000000001 or 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999199999999999977

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

   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 54.8

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

   7. Applied rewrites 54.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.1%

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.999199999999999977 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9992:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.998:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{t\_2}}{\sin ky}}\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;t\_3 \leq 0.9992:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)) -1.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.998)
     (/ (sin th) (/ (sqrt t_2) (sin ky)))
     (if (<= t_3 -0.1)
       t_1
       (if (<= t_3 0.005)
         (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
         (if (<= t_3 0.9992) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow((hypot(sin(kx), sin(ky)) / (sin(ky) * th)), -1.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.998) {
		tmp = sin(th) / (sqrt(t_2) / sin(ky));
	} else if (t_3 <= -0.1) {
		tmp = t_1;
	} else if (t_3 <= 0.005) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else if (t_3 <= 0.9992) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow((Math.hypot(Math.sin(kx), Math.sin(ky)) / (Math.sin(ky) * th)), -1.0);
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.998) {
		tmp = Math.sin(th) / (Math.sqrt(t_2) / Math.sin(ky));
	} else if (t_3 <= -0.1) {
		tmp = t_1;
	} else if (t_3 <= 0.005) {
		tmp = -ky * ((-1.0 / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th));
	} else if (t_3 <= 0.9992) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow((math.hypot(math.sin(kx), math.sin(ky)) / (math.sin(ky) * th)), -1.0)
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))
	tmp = 0
	if t_3 <= -0.998:
		tmp = math.sin(th) / (math.sqrt(t_2) / math.sin(ky))
	elif t_3 <= -0.1:
		tmp = t_1
	elif t_3 <= 0.005:
		tmp = -ky * ((-1.0 / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th))
	elif t_3 <= 0.9992:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)) ^ -1.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.998)
		tmp = Float64(sin(th) / Float64(sqrt(t_2) / sin(ky)));
	elseif (t_3 <= -0.1)
		tmp = t_1;
	elseif (t_3 <= 0.005)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * sin(th)));
	elseif (t_3 <= 0.9992)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (hypot(sin(kx), sin(ky)) / (sin(ky) * th)) ^ -1.0;
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_2));
	tmp = 0.0;
	if (t_3 <= -0.998)
		tmp = sin(th) / (sqrt(t_2) / sin(ky));
	elseif (t_3 <= -0.1)
		tmp = t_1;
	elseif (t_3 <= 0.005)
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	elseif (t_3 <= 0.9992)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.998], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$1, If[LessEqual[t$95$3, 0.005], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9992], t$95$1, 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))))) < -0.998

   1. Initial program 94.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 28.1

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 28.1

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.1

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

   10. Applied rewrites 93.1%

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

   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))))) < -0.10000000000000001 or 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999199999999999977

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

   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 54.8

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

   7. Applied rewrites 54.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.1%

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.999199999999999977 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.2%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 95.5

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

   5. Applied rewrites 95.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9992:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;t\_2 \leq 0.9992:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)) -1.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.1)
     t_1
     (if (<= t_2 0.005)
       (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
       (if (<= t_2 0.9992) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow((hypot(sin(kx), sin(ky)) / (sin(ky) * th)), -1.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.005) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else if (t_2 <= 0.9992) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow((Math.hypot(Math.sin(kx), Math.sin(ky)) / (Math.sin(ky) * th)), -1.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.005) {
		tmp = -ky * ((-1.0 / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th));
	} else if (t_2 <= 0.9992) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow((math.hypot(math.sin(kx), math.sin(ky)) / (math.sin(ky) * th)), -1.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.1:
		tmp = t_1
	elif t_2 <= 0.005:
		tmp = -ky * ((-1.0 / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th))
	elif t_2 <= 0.9992:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)) ^ -1.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.005)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * sin(th)));
	elseif (t_2 <= 0.9992)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (hypot(sin(kx), sin(ky)) / (sin(ky) * th)) ^ -1.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.005)
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	elseif (t_2 <= 0.9992)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -0.1], t$95$1, If[LessEqual[t$95$2, 0.005], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9992], 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))))) < -0.10000000000000001 or 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999199999999999977

   1. Initial program 97.1%

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

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 50.4

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

   7. Applied rewrites 50.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.1%

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.999199999999999977 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.2%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 95.5

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

   5. Applied rewrites 95.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9992:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;\frac{ky}{{t\_1}^{0.5}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.1)
     (* (/ (sin ky) (sqrt (- 1.0 (cos (* 2.0 kx))))) (sin th))
     (if (<= t_2 0.005) (* (/ ky (pow t_1 0.5)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (sin(ky) / sqrt((1.0 - cos((2.0 * kx))))) * sin(th);
	} else if (t_2 <= 0.005) {
		tmp = (ky / pow(t_1, 0.5)) * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(kx) ** 2.0d0
    t_2 = sin(ky) / sqrt((t_1 + (sin(ky) ** 2.0d0)))
    if (t_2 <= (-0.1d0)) then
        tmp = (sin(ky) / sqrt((1.0d0 - cos((2.0d0 * kx))))) * sin(th)
    else if (t_2 <= 0.005d0) then
        tmp = (ky / (t_1 ** 0.5d0)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (Math.sin(ky) / Math.sqrt((1.0 - Math.cos((2.0 * kx))))) * Math.sin(th);
	} else if (t_2 <= 0.005) {
		tmp = (ky / Math.pow(t_1, 0.5)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.1:
		tmp = (math.sin(ky) / math.sqrt((1.0 - math.cos((2.0 * kx))))) * math.sin(th)
	elif t_2 <= 0.005:
		tmp = (ky / math.pow(t_1, 0.5)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(1.0 - cos(Float64(2.0 * kx))))) * sin(th));
	elseif (t_2 <= 0.005)
		tmp = Float64(Float64(ky / (t_1 ^ 0.5)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = sin(ky) / sqrt((t_1 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.1)
		tmp = (sin(ky) / sqrt((1.0 - cos((2.0 * kx))))) * sin(th);
	elseif (t_2 <= 0.005)
		tmp = (ky / (t_1 ^ 0.5)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.005], N[(N[(ky / N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 96.4%

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

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 9.5%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.1%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 96.9%

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

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.0

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

   5. Applied rewrites 62.0%

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

Alternative 7: 45.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.1)
   (* (/ (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(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.1) {
		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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.1d0) 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.1) {
		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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.1:
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.1], 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))))) < 0.10000000000000001

   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 ky around 0

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

      1. lower-sin.f64 34.7

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

   5. Applied rewrites 34.7%

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

   if 0.10000000000000001 < (/.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 92.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 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. Add Preprocessing

Alternative 8: 45.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.1)
   (* (/ (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(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.1) {
		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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.1d0) 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.1) {
		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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.1:
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.1], 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))))) < 0.10000000000000001

   1. Initial program 97.8%

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

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

      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

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 34.7

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

   7. Applied rewrites 34.7%

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

   if 0.10000000000000001 < (/.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 92.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 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. Add Preprocessing

Alternative 9: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.005)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.005) {
		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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.005d0) 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.005) {
		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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.005:
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.005)
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.005)
		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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   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 ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.1

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

   5. Applied rewrites 33.1%

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

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 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. Add Preprocessing

Alternative 10: 36.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.005)
   (* (/ ky kx) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.005) {
		tmp = (ky / 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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.005d0) then
        tmp = (ky / 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.005) {
		tmp = (ky / 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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.005:
		tmp = (ky / 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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.005)
		tmp = Float64(Float64(ky / 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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.005)
		tmp = (ky / 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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(ky / kx), $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))))) < 0.0050000000000000001

   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 ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 20.3%

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

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 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. Add Preprocessing

Alternative 11: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5.8e-74)
   (* (pow th 3.0) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5.8e-74) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5.8d-74) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5.8e-74) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5.8e-74:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5.8e-74)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5.8e-74)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.8e-74], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.8e-74

   1. Initial program 97.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.5

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

   5. Applied rewrites 3.5%

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

   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 3.3%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 15.1%

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

   if 5.8e-74 < (/.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 93.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 54.8

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

   5. Applied rewrites 54.8%

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

Alternative 12: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (/ (sin ky) (sqrt (- 1.0 (cos (* 2.0 kx))))) (sin th))
   (if (<= (sin ky) 5e-6)
     (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) / sqrt((1.0 - cos((2.0 * kx))))) * sin(th);
	} else if (sin(ky) <= 5e-6) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = (Math.sin(ky) / Math.sqrt((1.0 - Math.cos((2.0 * kx))))) * Math.sin(th);
	} else if (Math.sin(ky) <= 5e-6) {
		tmp = -ky * ((-1.0 / Math.hypot(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) <= -0.02:
		tmp = (math.sin(ky) / math.sqrt((1.0 - math.cos((2.0 * kx))))) * math.sin(th)
	elif math.sin(ky) <= 5e-6:
		tmp = -ky * ((-1.0 / math.hypot(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 (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(1.0 - cos(Float64(2.0 * kx))))) * sin(th));
	elseif (sin(ky) <= 5e-6)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(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) <= -0.02)
		tmp = (sin(ky) / sqrt((1.0 - cos((2.0 * kx))))) * sin(th);
	elseif (sin(ky) <= 5e-6)
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-6], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 11.2%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 5.00000000000000041e-6

   1. Initial program 93.5%

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

Alternative 13: 49.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.035:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 0.5, ky \cdot ky\right)}}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-166}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.035)
   (/
    (sin th)
    (/ (sqrt (fma (- 1.0 (cos (* 2.0 kx))) 0.5 (* ky ky))) (sin ky)))
   (if (<= (sin kx) 4e-166)
     (sin th)
     (if (<= (sin kx) 5e-35)
       (* (/ (sin ky) (sqrt (+ (* kx kx) (* ky ky)))) (sin th))
       (* (/ (sin th) (sin kx)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.035) {
		tmp = sin(th) / (sqrt(fma((1.0 - cos((2.0 * kx))), 0.5, (ky * ky))) / sin(ky));
	} else if (sin(kx) <= 4e-166) {
		tmp = sin(th);
	} else if (sin(kx) <= 5e-35) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.035)
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 0.5, Float64(ky * ky))) / sin(ky)));
	elseif (sin(kx) <= 4e-166)
		tmp = sin(th);
	elseif (sin(kx) <= 5e-35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.035], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-166], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.035000000000000003

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 53.1

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

   5. Applied rewrites 53.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 53.2

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 53.2

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

   7. Applied rewrites 53.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-inverses N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\cos \color{blue}{0} - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}}{\sin ky}} \]

      9. cos-0 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. count-2 N/A

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

      13. lower-*.f64 53.0

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

   9. Applied rewrites 53.0%

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

   if -0.035000000000000003 < (sin.f64 kx) < 4.00000000000000016e-166

   1. Initial program 91.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 30.2

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

   5. Applied rewrites 30.2%

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

   if 4.00000000000000016e-166 < (sin.f64 kx) < 4.99999999999999964e-35

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.9

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

   8. Applied rewrites 57.9%

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

   if 4.99999999999999964e-35 < (sin.f64 kx)

   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. 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 99.4

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 59.1

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

   7. Applied rewrites 59.1%

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

Alternative 14: 49.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.035:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-166}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.035)
   (*
    (/ (sin ky) (sqrt (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (* ky ky))))
    (sin th))
   (if (<= (sin kx) 4e-166)
     (sin th)
     (if (<= (sin kx) 5e-35)
       (* (/ (sin ky) (sqrt (+ (* kx kx) (* ky ky)))) (sin th))
       (* (/ (sin th) (sin kx)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.035) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * sin(th);
	} else if (sin(kx) <= 4e-166) {
		tmp = sin(th);
	} else if (sin(kx) <= 5e-35) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = (sin(th) / sin(kx)) * 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) <= (-0.035d0)) then
        tmp = (sin(ky) / sqrt(((0.5d0 - (cos((2.0d0 * kx)) * 0.5d0)) + (ky * ky)))) * sin(th)
    else if (sin(kx) <= 4d-166) then
        tmp = sin(th)
    else if (sin(kx) <= 5d-35) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th)
    else
        tmp = (sin(th) / sin(kx)) * 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) <= -0.035) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * Math.sin(th);
	} else if (Math.sin(kx) <= 4e-166) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 5e-35) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.035:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * math.sin(th)
	elif math.sin(kx) <= 4e-166:
		tmp = math.sin(th)
	elif math.sin(kx) <= 5e-35:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (ky * ky)))) * math.sin(th)
	else:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.035)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(ky * ky)))) * sin(th));
	elseif (sin(kx) <= 4e-166)
		tmp = sin(th);
	elseif (sin(kx) <= 5e-35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.035)
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * sin(th);
	elseif (sin(kx) <= 4e-166)
		tmp = sin(th);
	elseif (sin(kx) <= 5e-35)
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	else
		tmp = (sin(th) / sin(kx)) * sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.035], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-166], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.035000000000000003

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 53.1

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

   5. Applied rewrites 53.1%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      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 52.8

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

   7. Applied rewrites 52.8%

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

   if -0.035000000000000003 < (sin.f64 kx) < 4.00000000000000016e-166

   1. Initial program 91.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 30.2

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

   5. Applied rewrites 30.2%

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

   if 4.00000000000000016e-166 < (sin.f64 kx) < 4.99999999999999964e-35

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.9

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

   8. Applied rewrites 57.9%

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

   if 4.99999999999999964e-35 < (sin.f64 kx)

   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. 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 99.4

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 59.1

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

   7. Applied rewrites 59.1%

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

Alternative 15: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.1e-5)
   (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
   (*
    (/
     (sin th)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
      2.0))
    (sin ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.1e-5) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else {
		tmp = (sin(th) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.1e-5)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * sin(th)));
	else
		tmp = Float64(Float64(sin(th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.1e-5], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.1e-5

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.5

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

   7. Applied rewrites 69.5%

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

   if 1.1e-5 < ky

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. 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 99.6

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

      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. Step-by-step derivation

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin ky \]

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}} \cdot \sin ky \]

      13. metadata-eval N/A

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

   6. Applied rewrites 99.2%

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

Alternative 16: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\left(-ky\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.1e-5)
   (* (- ky) (* (/ -1.0 (hypot (sin ky) (sin kx))) (sin th)))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* ky 2.0))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.1e-5) {
		tmp = -ky * ((-1.0 / hypot(sin(ky), sin(kx))) * sin(th));
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.1e-5)
		tmp = Float64(Float64(-ky) * Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * sin(th)));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.1e-5], N[((-ky) * N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.1e-5

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.5

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

   7. Applied rewrites 69.5%

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

   if 1.1e-5 < ky

   1. Initial program 99.6%

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

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

Alternative 17: 24.3% 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 96.4%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 19.5

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

5. Applied rewrites 19.5%

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

Alternative 18: 13.6% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (fma (* th th) -0.16666666666666666 1.0) th))

C

double code(double kx, double ky, double th) {
	return fma((th * th), -0.16666666666666666, 1.0) * th;
}

Julia

function code(kx, ky, th)
	return Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 19.5

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

5. Applied rewrites 19.5%

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

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 12.0%

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

10. Applied rewrites 12.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(th \cdot th\right) \cdot th, th\right) \]
11. Step-by-step derivation

12. Applied rewrites 12.0%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(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)))