Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 10.6s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 92.9

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

   8. lift-sqrt.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-hypot.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(-ky\right)\right) \cdot \sin th\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.01:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 0.5, t\_2\right)\right)}^{-1}}\\ \mathbf{elif}\;t\_3 \leq 0.15:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.999999999981754:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (/ -1.0 (hypot (sin kx) (sin ky))) (- ky)) (sin th)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.99)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.01)
       (*
        (* (sin ky) th)
        (sqrt (pow (fma (- 1.0 (cos (* 2.0 kx))) 0.5 t_2) -1.0)))
       (if (<= t_3 0.15)
         t_1
         (if (<= t_3 0.999999999981754)
           (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx))))
           (if (<= t_3 1.0) (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ((-1.0 / hypot(sin(kx), sin(ky))) * -ky) * sin(th);
	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 <= -0.99) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.01) {
		tmp = (sin(ky) * th) * sqrt(pow(fma((1.0 - cos((2.0 * kx))), 0.5, t_2), -1.0));
	} else if (t_3 <= 0.15) {
		tmp = t_1;
	} else if (t_3 <= 0.999999999981754) {
		tmp = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	} else if (t_3 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-1.0 / hypot(sin(kx), sin(ky))) * Float64(-ky)) * sin(th))
	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 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.01)
		tmp = Float64(Float64(sin(ky) * th) * sqrt((fma(Float64(1.0 - cos(Float64(2.0 * kx))), 0.5, t_2) ^ -1.0)));
	elseif (t_3 <= 0.15)
		tmp = t_1;
	elseif (t_3 <= 0.999999999981754)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))));
	elseif (t_3 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(-1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * (-ky)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[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, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.15], t$95$1, If[LessEqual[t$95$3, 0.999999999981754], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 81.9%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002

   1. Initial program 99.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 45.2

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

   7. Applied rewrites 45.2%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 89.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in ky around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 97.7

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

   12. Applied rewrites 97.7%

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

   if 0.149999999999999994 < (/.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))))) < 0.99999999998175404

   1. Initial program 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 97.5

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 39.3

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

   7. Applied rewrites 39.3%

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

   if 0.99999999998175404 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 0.5, {\sin ky}^{2}\right)\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\left(\frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(-ky\right)\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.999999999981754:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(-ky\right)\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(-ky\right)\right) \cdot \sin th\\ t_2 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.15:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.999999999981754:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (/ -1.0 (hypot (sin kx) (sin ky))) (- ky)) (sin th)))
        (t_2 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx)))))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -0.99)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.01)
       t_2
       (if (<= t_4 0.15)
         t_1
         (if (<= t_4 0.999999999981754)
           t_2
           (if (<= t_4 1.0) (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ((-1.0 / hypot(sin(kx), sin(ky))) * -ky) * sin(th);
	double t_2 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -0.99) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.01) {
		tmp = t_2;
	} else if (t_4 <= 0.15) {
		tmp = t_1;
	} else if (t_4 <= 0.999999999981754) {
		tmp = t_2;
	} else if (t_4 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ((-1.0 / Math.hypot(Math.sin(kx), Math.sin(ky))) * -ky) * Math.sin(th);
	double t_2 = (-th * Math.sin(ky)) * (-1.0 / Math.hypot(Math.sin(ky), Math.sin(kx)));
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -0.99) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_3))) * Math.sin(th);
	} else if (t_4 <= -0.01) {
		tmp = t_2;
	} else if (t_4 <= 0.15) {
		tmp = t_1;
	} else if (t_4 <= 0.999999999981754) {
		tmp = t_2;
	} else if (t_4 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ((-1.0 / math.hypot(math.sin(kx), math.sin(ky))) * -ky) * math.sin(th)
	t_2 = (-th * math.sin(ky)) * (-1.0 / math.hypot(math.sin(ky), math.sin(kx)))
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_3))
	tmp = 0
	if t_4 <= -0.99:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_3))) * math.sin(th)
	elif t_4 <= -0.01:
		tmp = t_2
	elif t_4 <= 0.15:
		tmp = t_1
	elif t_4 <= 0.999999999981754:
		tmp = t_2
	elif t_4 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-1.0 / hypot(sin(kx), sin(ky))) * Float64(-ky)) * sin(th))
	t_2 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.01)
		tmp = t_2;
	elseif (t_4 <= 0.15)
		tmp = t_1;
	elseif (t_4 <= 0.999999999981754)
		tmp = t_2;
	elseif (t_4 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ((-1.0 / hypot(sin(kx), sin(ky))) * -ky) * sin(th);
	t_2 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_3));
	tmp = 0.0;
	if (t_4 <= -0.99)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	elseif (t_4 <= -0.01)
		tmp = t_2;
	elseif (t_4 <= 0.15)
		tmp = t_1;
	elseif (t_4 <= 0.999999999981754)
		tmp = t_2;
	elseif (t_4 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(-1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * (-ky)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.01], t$95$2, If[LessEqual[t$95$4, 0.15], t$95$1, If[LessEqual[t$95$4, 0.999999999981754], t$95$2, If[LessEqual[t$95$4, 1.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 81.9%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 0.149999999999999994 < (/.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))))) < 0.99999999998175404

   1. Initial program 98.4%

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

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 98.3

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 41.9

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

   7. Applied rewrites 41.9%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 89.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in ky around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 97.7

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

   12. Applied rewrites 97.7%

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

   if 0.99999999998175404 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 75.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \left(\frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(-ky\right)\right) \cdot \sin th\\ \mathbf{if}\;t\_2 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.15:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.999999999981754:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx)))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (* (/ -1.0 (hypot (sin kx) (sin ky))) (- ky)) (sin th))))
   (if (<= t_2 -0.01)
     t_1
     (if (<= t_2 0.15)
       t_3
       (if (<= t_2 0.999999999981754) t_1 (if (<= t_2 1.0) (sin th) t_3))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = ((-1.0 / hypot(sin(kx), sin(ky))) * -ky) * sin(th);
	double tmp;
	if (t_2 <= -0.01) {
		tmp = t_1;
	} else if (t_2 <= 0.15) {
		tmp = t_3;
	} else if (t_2 <= 0.999999999981754) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (-th * Math.sin(ky)) * (-1.0 / Math.hypot(Math.sin(ky), Math.sin(kx)));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = ((-1.0 / Math.hypot(Math.sin(kx), Math.sin(ky))) * -ky) * Math.sin(th);
	double tmp;
	if (t_2 <= -0.01) {
		tmp = t_1;
	} else if (t_2 <= 0.15) {
		tmp = t_3;
	} else if (t_2 <= 0.999999999981754) {
		tmp = t_1;
	} else if (t_2 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (-th * math.sin(ky)) * (-1.0 / math.hypot(math.sin(ky), math.sin(kx)))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = ((-1.0 / math.hypot(math.sin(kx), math.sin(ky))) * -ky) * math.sin(th)
	tmp = 0
	if t_2 <= -0.01:
		tmp = t_1
	elif t_2 <= 0.15:
		tmp = t_3
	elif t_2 <= 0.999999999981754:
		tmp = t_1
	elif t_2 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = t_3
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(Float64(-1.0 / hypot(sin(kx), sin(ky))) * Float64(-ky)) * sin(th))
	tmp = 0.0
	if (t_2 <= -0.01)
		tmp = t_1;
	elseif (t_2 <= 0.15)
		tmp = t_3;
	elseif (t_2 <= 0.999999999981754)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = ((-1.0 / hypot(sin(kx), sin(ky))) * -ky) * sin(th);
	tmp = 0.0;
	if (t_2 <= -0.01)
		tmp = t_1;
	elseif (t_2 <= 0.15)
		tmp = t_3;
	elseif (t_2 <= 0.999999999981754)
		tmp = t_1;
	elseif (t_2 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * (-ky)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.01], t$95$1, If[LessEqual[t$95$2, 0.15], t$95$3, If[LessEqual[t$95$2, 0.999999999981754], t$95$1, If[LessEqual[t$95$2, 1.0], N[Sin[th], $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 0.149999999999999994 < (/.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))))) < 0.99999999998175404

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 89.2

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 40.2

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

   7. Applied rewrites 40.2%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 89.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 95.1

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

   7. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in ky around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 97.7

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

   12. Applied rewrites 97.7%

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

   if 0.99999999998175404 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_2 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999999999981754:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx)))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.01)
     t_1
     (if (<= t_3 5e-8)
       (*
        (/
         (*
          (fma
           (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (sqrt (+ t_2 (* ky ky))))
        (sin th))
       (if (<= t_3 0.999999999981754) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.01) {
		tmp = t_1;
	} else if (t_3 <= 5e-8) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	} else if (t_3 <= 0.999999999981754) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.01)
		tmp = t_1;
	elseif (t_3 <= 5e-8)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_2 + Float64(ky * ky)))) * sin(th));
	elseif (t_3 <= 0.999999999981754)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.01], t$95$1, If[LessEqual[t$95$3, 5e-8], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999999999981754], 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))))) < -0.0100000000000000002 or 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999999998175404

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 89.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 40.4

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

   7. Applied rewrites 40.4%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      12. lower-*.f64 98.8

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

   if 0.99999999998175404 < (/.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 87.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 96.6

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

   5. Applied rewrites 96.6%

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

Alternative 6: 58.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_1 + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.01)
     (* (/ (sin ky) (* (sqrt 0.5) (sqrt (- 1.0 (cos (* 2.0 kx)))))) (sin th))
     (if (<= t_2 5e-8)
       (*
        (/
         (*
          (fma
           (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (sqrt (+ t_1 (* ky ky))))
        (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 <= -0.01) {
		tmp = (sin(ky) / (sqrt(0.5) * sqrt((1.0 - cos((2.0 * kx)))))) * sin(th);
	} else if (t_2 <= 5e-8) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_1 + (ky * ky)))) * sin(th);
	} else {
		tmp = 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 <= -0.01)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(0.5) * sqrt(Float64(1.0 - cos(Float64(2.0 * kx)))))) * sin(th));
	elseif (t_2 <= 5e-8)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_1 + Float64(ky * ky)))) * sin(th));
	else
		tmp = sin(th);
	end
	return 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, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-8], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $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))))) < -0.0100000000000000002

   1. Initial program 88.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 9.8

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

   7. Applied rewrites 9.8%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

      12. lower-*.f64 98.8

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 7: 58.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_1 + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.01)
     (* (/ (sin ky) (* (sqrt 0.5) (sqrt (- 1.0 (cos (* 2.0 kx)))))) (sin th))
     (if (<= t_2 5e-8)
       (*
        (/
         (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
         (sqrt (+ t_1 (* ky ky))))
        (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 <= -0.01) {
		tmp = (sin(ky) / (sqrt(0.5) * sqrt((1.0 - cos((2.0 * kx)))))) * sin(th);
	} else if (t_2 <= 5e-8) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / sqrt((t_1 + (ky * ky)))) * sin(th);
	} else {
		tmp = 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 <= -0.01)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(0.5) * sqrt(Float64(1.0 - cos(Float64(2.0 * kx)))))) * sin(th));
	elseif (t_2 <= 5e-8)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_1 + Float64(ky * ky)))) * sin(th));
	else
		tmp = sin(th);
	end
	return 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, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-8], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $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))))) < -0.0100000000000000002

   1. Initial program 88.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 9.8

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

   7. Applied rewrites 9.8%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th \]

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 8: 48.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 2e-184)
     (* (/ (sin ky) (* (sqrt 0.5) (sqrt (- 1.0 (cos (* 2.0 kx)))))) (sin th))
     (if (<= t_1 5e-8) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-184) {
		tmp = (sin(ky) / (sqrt(0.5) * sqrt((1.0 - cos((2.0 * kx)))))) * sin(th);
	} else if (t_1 <= 5e-8) {
		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) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 2d-184) then
        tmp = (sin(ky) / (sqrt(0.5d0) * sqrt((1.0d0 - cos((2.0d0 * kx)))))) * sin(th)
    else if (t_1 <= 5d-8) 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 t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-184) {
		tmp = (Math.sin(ky) / (Math.sqrt(0.5) * Math.sqrt((1.0 - Math.cos((2.0 * kx)))))) * Math.sin(th);
	} else if (t_1 <= 5e-8) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 2e-184:
		tmp = (math.sin(ky) / (math.sqrt(0.5) * math.sqrt((1.0 - math.cos((2.0 * kx)))))) * math.sin(th)
	elif t_1 <= 5e-8:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-184)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(0.5) * sqrt(Float64(1.0 - cos(Float64(2.0 * kx)))))) * sin(th));
	elseif (t_1 <= 5e-8)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 2e-184)
		tmp = (sin(ky) / (sqrt(0.5) * sqrt((1.0 - cos((2.0 * kx)))))) * sin(th);
	elseif (t_1 <= 5e-8)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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$1, 2e-184], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-8], N[(N[(ky / N[Sin[kx], $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))))) < 2.0000000000000001e-184

   1. Initial program 92.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cos-diff N/A

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

      10. cos-sin-sum N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 80.9

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 34.0

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

   7. Applied rewrites 34.0%

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

   if 2.0000000000000001e-184 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   1. Initial program 97.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 9: 45.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-8)
   (* (/ (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)))) <= 5e-8) {
		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)))) <= 5d-8) 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)))) <= 5e-8) {
		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)))) <= 5e-8:
		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)))) <= 5e-8)
		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)))) <= 5e-8)
		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], 5e-8], 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))))) < 4.9999999999999998e-8

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 38.3

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

   5. Applied rewrites 38.3%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 10: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-8)
   (* (/ 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)))) <= 5e-8) {
		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)))) <= 5d-8) 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)))) <= 5e-8) {
		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)))) <= 5e-8:
		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)))) <= 5e-8)
		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)))) <= 5e-8)
		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], 5e-8], 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))))) < 4.9999999999999998e-8

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 36.5

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

   5. Applied rewrites 36.5%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 11: 31.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{0.5}{ky \cdot ky}, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-62)
   (/ (sin th) (fma (/ 0.5 (* ky ky)) (* kx kx) 1.0))
   (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)))) <= 5e-62) {
		tmp = sin(th) / fma((0.5 / (ky * ky)), (kx * kx), 1.0);
	} else {
		tmp = 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)))) <= 5e-62)
		tmp = Float64(sin(th) / fma(Float64(0.5 / Float64(ky * ky)), Float64(kx * kx), 1.0));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-62], N[(N[Sin[th], $MachinePrecision] / N[(N[(0.5 / N[(ky * ky), $MachinePrecision]), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000002e-62

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 18.0

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

   7. Applied rewrites 18.0%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 17.8%

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

   if 5.0000000000000002e-62 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 63.4

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

   5. Applied rewrites 63.4%

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

Alternative 12: 15.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-302}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-302)
   (* (* (* -0.16666666666666666 th) th) th)
   (*
    (fma
     (fma (* th th) 0.008333333333333333 -0.16666666666666666)
     (* th th)
     1.0)
    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)))) * sin(th)) <= 1e-302) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = fma(fma((th * th), 0.008333333333333333, -0.16666666666666666), (th * th), 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-302)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(fma(fma(Float64(th * th), 0.008333333333333333, -0.16666666666666666), Float64(th * th), 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[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], 1e-302], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.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))))) (sin.f64 th)) < 9.9999999999999996e-303

   1. Initial program 94.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 25.5

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

   5. Applied rewrites 25.5%

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

   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 13.6%

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

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 19.1%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 19.1%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 9.9999999999999996e-303 < (*.f64 (/.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))))) (sin.f64 th))

   1. Initial program 91.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 28.0

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

   5. Applied rewrites 28.0%

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

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 15.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-302}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-302)
   (* (* (* -0.16666666666666666 th) th) th)
   (* (fma (* -0.16666666666666666 th) th 1.0) 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)))) * sin(th)) <= 1e-302) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = fma((-0.16666666666666666 * th), th, 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-302)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * th), th, 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[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], 1e-302], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th + 1.0), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.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))))) (sin.f64 th)) < 9.9999999999999996e-303

   1. Initial program 94.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 25.5

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

   5. Applied rewrites 25.5%

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

   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 13.6%

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

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 19.1%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 19.1%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 9.9999999999999996e-303 < (*.f64 (/.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))))) (sin.f64 th))

   1. Initial program 91.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 28.0

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

   5. Applied rewrites 28.0%

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

   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 14.9%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.9%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-62)
   (* (* (* -0.16666666666666666 th) th) 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)))) <= 5e-62) {
		tmp = ((-0.16666666666666666 * th) * th) * 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)))) <= 5d-62) then
        tmp = (((-0.16666666666666666d0) * th) * th) * 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)))) <= 5e-62) {
		tmp = ((-0.16666666666666666 * th) * th) * 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)))) <= 5e-62:
		tmp = ((-0.16666666666666666 * th) * th) * 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)))) <= 5e-62)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * 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)))) <= 5e-62)
		tmp = ((-0.16666666666666666 * th) * th) * 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], 5e-62], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000002e-62

   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 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 3.4%

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

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 16.9%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 16.9%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 5.0000000000000002e-62 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 63.4

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

   5. Applied rewrites 63.4%

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

Alternative 15: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 92.9

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

   8. lift-sqrt.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-hypot.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 16: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(-ky\right)\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 5.4e-6)
   (* (* (/ -1.0 (hypot (sin kx) (sin ky))) (- ky)) (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 <= 5.4e-6) {
		tmp = ((-1.0 / hypot(sin(kx), sin(ky))) * -ky) * 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 <= 5.4e-6)
		tmp = Float64(Float64(Float64(-1.0 / hypot(sin(kx), sin(ky))) * Float64(-ky)) * 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, 5.4e-6], N[(N[(N[(-1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * (-ky)), $MachinePrecision] * N[Sin[th], $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 < 5.39999999999999997e-6

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 87.6

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 66.2%

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

   10. Taylor expanded in ky around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 66.5

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

   12. Applied rewrites 66.5%

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

   if 5.39999999999999997e-6 < ky

   1. Initial program 99.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.2%

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

Alternative 17: 11.0% accurate, 39.5× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	return ((-0.16666666666666666 * th) * th) * 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 = (((-0.16666666666666666d0) * th) * th) * th
end function

Java

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

Python

def code(kx, ky, th):
	return ((-0.16666666666666666 * th) * th) * th

Julia

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

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((-0.16666666666666666 * th) * th) * th;
end

Wolfram

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

Derivation

1. Initial program 93.0%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 26.7

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

5. Applied rewrites 26.7%

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

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 14.2%

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

9. Taylor expanded in th around inf

   \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 11.8%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
12. Step-by-step derivation

13. Applied rewrites 11.8%

    \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024303 
(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)))