Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 15.5s

Alternatives: 26

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. accelerator-lowering-hypot.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. sin-lowering-sin.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), 1\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 2e-5)
     (*
      (sin th)
      (/
       (sin ky)
       (hypot
        (sin ky)
        (*
         kx
         (fma
          (* kx kx)
          (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
          1.0)))))
     (if (<= t_1 0.12)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (/ (sin th) (/ (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5)) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma((kx * kx), fma(0.008333333333333333, (kx * kx), -0.16666666666666666), 1.0))));
	} else if (t_1 <= 0.12) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(th) / (sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(Float64(kx * kx), fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), 1.0)))));
	elseif (t_1 <= 0.12)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(th) / Float64(sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5)) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(N[(kx * kx), $MachinePrecision] * N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000016e-5

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.8

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

   7. Simplified 99.8%

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

   if 2.00000000000000016e-5 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.12

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 67.7%

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

   if 0.12 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 59.3

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

   7. Simplified 59.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), 1\right)\right)}\\ \mathbf{elif}\;{\sin kx}^{2} \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 2e-5)
     (*
      (sin th)
      (/
       (sin ky)
       (hypot (sin ky) (* kx (fma -0.16666666666666666 (* kx kx) 1.0)))))
     (if (<= t_1 0.12)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (/ (sin th) (/ (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5)) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma(-0.16666666666666666, (kx * kx), 1.0))));
	} else if (t_1 <= 0.12) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(th) / (sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(-0.16666666666666666, Float64(kx * kx), 1.0)))));
	elseif (t_1 <= 0.12)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(th) / Float64(sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5)) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000016e-5

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 99.8

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

   7. Simplified 99.8%

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

   if 2.00000000000000016e-5 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.12

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 67.7%

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

   if 0.12 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 59.3

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

   7. Simplified 59.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{elif}\;{\sin kx}^{2} \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 2e-5)
     (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
     (if (<= t_1 0.12)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (/ (sin th) (/ (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5)) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (t_1 <= 0.12) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(th) / (sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (t_1 <= 0.12)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(th) / Float64(sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5)) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000016e-5

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 99.5%

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

   if 2.00000000000000016e-5 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.12

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 67.7%

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

   if 0.12 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 59.3

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

   7. Simplified 59.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;{\sin kx}^{2} \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-164}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{elif}\;\sin ky \leq 10^{-54}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, kx \cdot kx\right)}}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (* (sin ky) th) (sqrt (/ 2.0 (- 1.0 (cos (* ky -2.0))))))
   (if (<= (sin ky) 5e-164)
     (* (sin th) (* ky (/ 1.0 (sin kx))))
     (if (<= (sin ky) 1e-54)
       (/
        (sin th)
        (/
         (sqrt
          (fma
           (*
            (* ky ky)
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma (* ky ky) -0.006349206349206349 0.08888888888888889)
              -0.6666666666666666)
             2.0))
           0.5
           (* kx kx)))
         (sin ky)))
       (if (<= (sin ky) 2e-23)
         (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) * th) * sqrt((2.0 / (1.0 - cos((ky * -2.0)))));
	} else if (sin(ky) <= 5e-164) {
		tmp = sin(th) * (ky * (1.0 / sin(kx)));
	} else if (sin(ky) <= 1e-54) {
		tmp = sin(th) / (sqrt(fma(((ky * ky) * fma((ky * ky), fma((ky * ky), fma((ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, (kx * kx))) / sin(ky));
	} else if (sin(ky) <= 2e-23) {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(ky * -2.0))))));
	elseif (sin(ky) <= 5e-164)
		tmp = Float64(sin(th) * Float64(ky * Float64(1.0 / sin(kx))));
	elseif (sin(ky) <= 1e-54)
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(Float64(ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, Float64(kx * kx))) / sin(ky)));
	elseif (sin(ky) <= 2e-23)
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-164], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-54], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-23], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   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. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 53.9%

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

   8. Taylor expanded in kx around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 30.4

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

   10. Simplified 30.4%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 4.99999999999999962e-164

   1. Initial program 82.9%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 40.6%

      \[\leadsto \color{blue}{\left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-0.5}{{\sin kx}^{3}} + \frac{-0.16666666666666666}{\sin kx}, \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

   6. Taylor expanded in ky around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 43.5

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

   8. Simplified 43.5%

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

   if 4.99999999999999962e-164 < (sin.f64 ky) < 1e-54

   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. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 57.2%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right) + 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \color{blue}{\frac{-2}{3}}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-2}{315} \cdot {ky}^{2} + \frac{4}{45}}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-2}{315}} + \frac{4}{45}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-2}{315}, \frac{4}{45}\right)}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      17. *-lowering-*.f64 94.0

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   7. Simplified 94.0%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}}{\sin ky}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}}{\sin ky}} \]

      2. *-lowering-*.f64 53.4

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, \color{blue}{kx \cdot kx}\right)}}{\sin ky}} \]

   10. Simplified 53.4%

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, \color{blue}{kx \cdot kx}\right)}}{\sin ky}} \]

   if 1e-54 < (sin.f64 ky) < 1.99999999999999992e-23

   1. Initial program 98.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 49.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. cos-neg N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 49.3

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

   7. Simplified 49.3%

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

   if 1.99999999999999992e-23 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 59.6

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

   5. Simplified 59.6%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-164}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{elif}\;\sin ky \leq 10^{-54}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, kx \cdot kx\right)}}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-5)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot
      (sin ky)
      (*
       kx
       (fma
        (* kx kx)
        (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
        1.0)))))
   (*
    (sin th)
    (*
     (sin ky)
     (sqrt
      (/
       1.0
       (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx)))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-5) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma((kx * kx), fma(0.008333333333333333, (kx * kx), -0.16666666666666666), 1.0))));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx))))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(Float64(kx * kx), fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), 1.0)))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(N[(kx * kx), $MachinePrecision] * N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000016e-5

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.8

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

   7. Simplified 99.8%

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

   if 2.00000000000000016e-5 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-5)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot
      (sin ky)
      (*
       kx
       (fma
        (* kx kx)
        (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
        1.0)))))
   (/
    (* (sin ky) (sin th))
    (sqrt
     (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-5) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma((kx * kx), fma(0.008333333333333333, (kx * kx), -0.16666666666666666), 1.0))));
	} else {
		tmp = (sin(ky) * sin(th)) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(Float64(kx * kx), fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), 1.0)))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(N[(kx * kx), $MachinePrecision] * N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000016e-5

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.8

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

   7. Simplified 99.8%

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

   if 2.00000000000000016e-5 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. sin-mult N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-5)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot
      (sin ky)
      (*
       kx
       (fma
        (* kx kx)
        (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
        1.0)))))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-5) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma((kx * kx), fma(0.008333333333333333, (kx * kx), -0.16666666666666666), 1.0))));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(Float64(kx * kx), fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), 1.0)))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(N[(kx * kx), $MachinePrecision] * N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000016e-5

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.8

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

   7. Simplified 99.8%

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

   if 2.00000000000000016e-5 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. count-2 N/A

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

      9. +-inverses N/A

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

      10. cos-0 N/A

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

      11. --lowering--.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a N/A

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

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

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

      18. +-lowering-+.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. count-2 N/A

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

      22. cos-lowering-cos.f64 N/A

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

      23. +-lowering-+.f64 99.2

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (* (sin ky) th) (sqrt (/ 2.0 (- 1.0 (cos (* ky -2.0))))))
   (if (<= (sin ky) 2e-170)
     (* (sin th) (/ (sin ky) (sin kx)))
     (if (<= (sin ky) 1e-7)
       (/
        (sin th)
        (/
         (sqrt
          (fma
           (*
            (* ky ky)
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma (* ky ky) -0.006349206349206349 0.08888888888888889)
              -0.6666666666666666)
             2.0))
           0.5
           (+ 0.5 (* -0.5 (cos (+ kx kx))))))
         (* ky (fma (* ky ky) -0.16666666666666666 1.0))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) * th) * sqrt((2.0 / (1.0 - cos((ky * -2.0)))));
	} else if (sin(ky) <= 2e-170) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (sin(ky) <= 1e-7) {
		tmp = sin(th) / (sqrt(fma(((ky * ky) * fma((ky * ky), fma((ky * ky), fma((ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))) / (ky * fma((ky * ky), -0.16666666666666666, 1.0)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(ky * -2.0))))));
	elseif (sin(ky) <= 2e-170)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (sin(ky) <= 1e-7)
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(Float64(ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))))) / Float64(ky * fma(Float64(ky * ky), -0.16666666666666666, 1.0))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-170], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   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. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 53.9%

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

   8. Taylor expanded in kx around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 30.4

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

   10. Simplified 30.4%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 1.99999999999999997e-170

   1. Initial program 82.9%

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

   3. Taylor expanded in ky around 0

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

      1. sin-lowering-sin.f64 43.6

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

   5. Simplified 43.6%

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

   if 1.99999999999999997e-170 < (sin.f64 ky) < 9.9999999999999995e-8

   1. Initial program 99.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 50.8%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right) + 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \color{blue}{\frac{-2}{3}}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-2}{315} \cdot {ky}^{2} + \frac{4}{45}}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-2}{315}} + \frac{4}{45}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-2}{315}, \frac{4}{45}\right)}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      17. *-lowering-*.f64 93.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   7. Simplified 93.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right)}} \]

      6. *-lowering-*.f64 93.4

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right)}} \]

   10. Simplified 93.4%

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}}} \]

   if 9.9999999999999995e-8 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 58.4

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

   5. Simplified 58.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (* (sin ky) th) (sqrt (/ 2.0 (- 1.0 (cos (* ky -2.0))))))
   (if (<= (sin ky) 2e-170)
     (* (sin th) (/ ky (sin kx)))
     (if (<= (sin ky) 1e-7)
       (/
        (sin th)
        (/
         (sqrt
          (fma
           (*
            (* ky ky)
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma (* ky ky) -0.006349206349206349 0.08888888888888889)
              -0.6666666666666666)
             2.0))
           0.5
           (+ 0.5 (* -0.5 (cos (+ kx kx))))))
         (* ky (fma (* ky ky) -0.16666666666666666 1.0))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) * th) * sqrt((2.0 / (1.0 - cos((ky * -2.0)))));
	} else if (sin(ky) <= 2e-170) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (sin(ky) <= 1e-7) {
		tmp = sin(th) / (sqrt(fma(((ky * ky) * fma((ky * ky), fma((ky * ky), fma((ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))) / (ky * fma((ky * ky), -0.16666666666666666, 1.0)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(ky * -2.0))))));
	elseif (sin(ky) <= 2e-170)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (sin(ky) <= 1e-7)
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(Float64(ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))))) / Float64(ky * fma(Float64(ky * ky), -0.16666666666666666, 1.0))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-170], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   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. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 53.9%

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

   8. Taylor expanded in kx around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 30.4

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

   10. Simplified 30.4%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 1.99999999999999997e-170

   1. Initial program 82.9%

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

   3. Taylor expanded in ky around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 43.5

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

   5. Simplified 43.5%

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

   if 1.99999999999999997e-170 < (sin.f64 ky) < 9.9999999999999995e-8

   1. Initial program 99.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 50.8%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right) + 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \color{blue}{\frac{-2}{3}}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-2}{315} \cdot {ky}^{2} + \frac{4}{45}}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-2}{315}} + \frac{4}{45}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-2}{315}, \frac{4}{45}\right)}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      17. *-lowering-*.f64 93.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   7. Simplified 93.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right)}} \]

      6. *-lowering-*.f64 93.4

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right)}} \]

   10. Simplified 93.4%

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}}} \]

   if 9.9999999999999995e-8 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 58.4

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

   5. Simplified 58.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (* (sin ky) th) (sqrt (/ 2.0 (- 1.0 (cos (* ky -2.0))))))
   (if (<= (sin ky) 2e-170)
     (* (sin th) (/ ky (sin kx)))
     (if (<= (sin ky) 1e-7)
       (/
        (sin th)
        (/
         (sqrt
          (fma
           (*
            (* ky ky)
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma (* ky ky) -0.006349206349206349 0.08888888888888889)
              -0.6666666666666666)
             2.0))
           0.5
           (+ 0.5 (* -0.5 (cos (+ kx kx))))))
         ky))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) * th) * sqrt((2.0 / (1.0 - cos((ky * -2.0)))));
	} else if (sin(ky) <= 2e-170) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (sin(ky) <= 1e-7) {
		tmp = sin(th) / (sqrt(fma(((ky * ky) * fma((ky * ky), fma((ky * ky), fma((ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(ky * -2.0))))));
	elseif (sin(ky) <= 2e-170)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (sin(ky) <= 1e-7)
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(Float64(ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))))) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-170], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   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. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 53.9%

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

   8. Taylor expanded in kx around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 30.4

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

   10. Simplified 30.4%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 1.99999999999999997e-170

   1. Initial program 82.9%

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

   3. Taylor expanded in ky around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 43.5

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

   5. Simplified 43.5%

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

   if 1.99999999999999997e-170 < (sin.f64 ky) < 9.9999999999999995e-8

   1. Initial program 99.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 50.8%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right) + 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \color{blue}{\frac{-2}{3}}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-2}{315} \cdot {ky}^{2} + \frac{4}{45}}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-2}{315}} + \frac{4}{45}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-2}{315}, \frac{4}{45}\right)}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      17. *-lowering-*.f64 93.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   7. Simplified 93.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky}}} \]
   9. Step-by-step derivation

   10. Simplified 93.4%

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky}}} \]

   if 9.9999999999999995e-8 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 58.4

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

   5. Simplified 58.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-184}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (* (sin ky) th) (sqrt (/ 2.0 (- 1.0 (cos (* ky -2.0))))))
   (if (<= (sin ky) 5e-184) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) * th) * sqrt((2.0 / (1.0 - cos((ky * -2.0)))));
	} else if (sin(ky) <= 5e-184) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = (sin(ky) * th) * sqrt((2.0d0 / (1.0d0 - cos((ky * (-2.0d0))))))
    else if (sin(ky) <= 5d-184) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = (Math.sin(ky) * th) * Math.sqrt((2.0 / (1.0 - Math.cos((ky * -2.0)))));
	} else if (Math.sin(ky) <= 5e-184) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = (math.sin(ky) * th) * math.sqrt((2.0 / (1.0 - math.cos((ky * -2.0)))))
	elif math.sin(ky) <= 5e-184:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(ky * -2.0))))));
	elseif (sin(ky) <= 5e-184)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = (sin(ky) * th) * sqrt((2.0 / (1.0 - cos((ky * -2.0)))));
	elseif (sin(ky) <= 5e-184)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-184], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 53.9%

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

   8. Taylor expanded in kx around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 30.4

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

   10. Simplified 30.4%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 5.00000000000000003e-184

   1. Initial program 84.1%

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

   3. Taylor expanded in ky around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 44.1

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

   5. Simplified 44.1%

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

   if 5.00000000000000003e-184 < (sin.f64 ky)

   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 kx around 0

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

      1. sin-lowering-sin.f64 55.1

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

   5. Simplified 55.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-184}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.12:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.12)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (/ (sin th) (/ (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5)) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.12) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) / (sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.12)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) / Float64(sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5)) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.12], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.1200000000000001

   1. Initial program 90.6%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in kx around 0

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

   7. Simplified 71.4%

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

   if 1.1200000000000001 < kx

   1. Initial program 99.4%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 59.7

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

   7. Simplified 59.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.12:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.4:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.4)
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
   (/ (sin th) (/ (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0))))) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.4) {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th) / (sqrt((0.5 * (1.0 - cos((ky * -2.0))))) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.4) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th) / (Math.sqrt((0.5 * (1.0 - Math.cos((ky * -2.0))))) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.4:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th) / (math.sqrt((0.5 * (1.0 - math.cos((ky * -2.0))))) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.4)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(th) / Float64(sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0))))) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.4)
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	else
		tmp = sin(th) / (sqrt((0.5 * (1.0 - cos((ky * -2.0))))) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.4], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.40000000000000002

   1. Initial program 90.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Simplified 68.0%

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

   if 0.40000000000000002 < ky

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. --lowering--.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 50.5

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

   7. Simplified 50.5%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}}{\sin ky}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.4:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.92 \cdot 10^{-168}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 0.44:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.92e-168)
   (* (sin th) (/ ky (sin kx)))
   (if (<= ky 0.44)
     (/
      (sin th)
      (/
       (sqrt
        (fma
         (*
          (* ky ky)
          (fma
           (* ky ky)
           (fma
            (* ky ky)
            (fma (* ky ky) -0.006349206349206349 0.08888888888888889)
            -0.6666666666666666)
           2.0))
         0.5
         (+ 0.5 (* -0.5 (cos (+ kx kx))))))
       (*
        ky
        (fma
         (* ky ky)
         (fma
          (* ky ky)
          (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666)
         1.0))))
     (/ (sin th) (/ (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0))))) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.92e-168) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (ky <= 0.44) {
		tmp = sin(th) / (sqrt(fma(((ky * ky) * fma((ky * ky), fma((ky * ky), fma((ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, (0.5 + (-0.5 * cos((kx + kx)))))) / (ky * fma((ky * ky), fma((ky * ky), fma((ky * ky), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0)));
	} else {
		tmp = sin(th) / (sqrt((0.5 * (1.0 - cos((ky * -2.0))))) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.92e-168)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (ky <= 0.44)
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(Float64(ky * ky), -0.006349206349206349, 0.08888888888888889), -0.6666666666666666), 2.0)), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))))) / Float64(ky * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0))));
	else
		tmp = Float64(sin(th) / Float64(sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0))))) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.92e-168], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.44], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.92000000000000009e-168

   1. Initial program 89.2%

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

   3. Taylor expanded in ky around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 28.1

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

   5. Simplified 28.1%

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

   if 1.92000000000000009e-168 < ky < 0.440000000000000002

   1. Initial program 99.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 50.8%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right) + 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \color{blue}{\frac{-2}{3}}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-2}{315} \cdot {ky}^{2} + \frac{4}{45}}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-2}{315}} + \frac{4}{45}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-2}{315}, \frac{4}{45}\right)}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      17. *-lowering-*.f64 93.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   7. Simplified 93.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right) + 1\right)}}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}, 1\right)}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}, 1\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}, 1\right)}} \]

      6. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right)}, 1\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), 1\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), 1\right)}} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-1}{5040} \cdot {ky}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)}} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)}} \]

      15. *-lowering-*.f64 93.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}} \]

   10. Simplified 93.4%

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\color{blue}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}}} \]

   if 0.440000000000000002 < ky

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. --lowering--.f64 N/A

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

      5. cos-neg N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 50.5

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

   7. Simplified 50.5%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}}{\sin ky}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.92 \cdot 10^{-168}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 0.44:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-184}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 5e-184) (* (sin ky) (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 5e-184) {
		tmp = sin(ky) * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 5d-184) then
        tmp = sin(ky) * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 5e-184) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 5e-184:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 5e-184)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 5e-184)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-184], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 5.00000000000000003e-184

   1. Initial program 90.3%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 60.4

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

   7. Simplified 60.4%

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

   8. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 18.3

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

   10. Simplified 18.3%

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

   11. Taylor expanded in kx around 0

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

       1. associate-/l* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. sin-lowering-sin.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. sin-lowering-sin.f64 18.4

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

   13. Simplified 18.4%

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

   if 5.00000000000000003e-184 < (sin.f64 ky)

   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 kx around 0

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

      1. sin-lowering-sin.f64 55.1

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

   5. Simplified 55.1%

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

Alternative 17: 35.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.2 \cdot 10^{-45}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.12:\\ \;\;\;\;\frac{\left(\sin ky \cdot \sin th\right) \cdot \mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.2e-45)
   (sin th)
   (if (<= kx 1.12)
     (/ (* (* (sin ky) (sin th)) (fma 0.16666666666666666 (* kx kx) 1.0)) kx)
     (*
      (sin th)
      (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.2e-45) {
		tmp = sin(th);
	} else if (kx <= 1.12) {
		tmp = ((sin(ky) * sin(th)) * fma(0.16666666666666666, (kx * kx), 1.0)) / kx;
	} else {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.2e-45)
		tmp = sin(th);
	elseif (kx <= 1.12)
		tmp = Float64(Float64(Float64(sin(ky) * sin(th)) * fma(0.16666666666666666, Float64(kx * kx), 1.0)) / kx);
	else
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.2e-45], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.12], N[(N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.19999999999999993e-45

   1. Initial program 90.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 26.3

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

   5. Simplified 26.3%

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

   if 2.19999999999999993e-45 < kx < 1.1200000000000001

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.4

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

   7. Simplified 85.4%

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

   8. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 51.9

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

   10. Simplified 51.9%

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

   11. Taylor expanded in kx around 0

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

       1. /-lowering-/.f64 N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. *-lowering-*.f64 N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. sin-lowering-sin.f64 N/A

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

       11. sin-lowering-sin.f64 52.1

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

   13. Simplified 52.1%

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

   if 1.1200000000000001 < kx

   1. Initial program 99.4%

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

      1. unpow2 N/A

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

      2. sin-mult N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. count-2 N/A

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

      7. cos-diff N/A

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

      8. cos-sin-sum N/A

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

      9. --lowering--.f64 N/A

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

      10. count-2 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. +-lowering-+.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. --lowering--.f64 N/A

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

      8. cos-neg N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sqrt-lowering-sqrt.f64 51.4

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

   7. Simplified 51.4%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.2 \cdot 10^{-45}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.12:\\ \;\;\;\;\frac{\left(\sin ky \cdot \sin th\right) \cdot \mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.25 \cdot 10^{-45}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.12:\\ \;\;\;\;\frac{\left(\sin ky \cdot \sin th\right) \cdot \mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\cos \left(kx \cdot -2\right), -0.5, 0.5\right)}}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.25e-45)
   (sin th)
   (if (<= kx 1.12)
     (/ (* (* (sin ky) (sin th)) (fma 0.16666666666666666 (* kx kx) 1.0)) kx)
     (/ (sin th) (/ (sqrt (fma (cos (* kx -2.0)) -0.5 0.5)) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.25e-45) {
		tmp = sin(th);
	} else if (kx <= 1.12) {
		tmp = ((sin(ky) * sin(th)) * fma(0.16666666666666666, (kx * kx), 1.0)) / kx;
	} else {
		tmp = sin(th) / (sqrt(fma(cos((kx * -2.0)), -0.5, 0.5)) / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.25e-45)
		tmp = sin(th);
	elseif (kx <= 1.12)
		tmp = Float64(Float64(Float64(sin(ky) * sin(th)) * fma(0.16666666666666666, Float64(kx * kx), 1.0)) / kx);
	else
		tmp = Float64(sin(th) / Float64(sqrt(fma(cos(Float64(kx * -2.0)), -0.5, 0.5)) / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.25e-45], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.12], N[(N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.2499999999999999e-45

   1. Initial program 90.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 26.3

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

   5. Simplified 26.3%

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

   if 2.2499999999999999e-45 < kx < 1.1200000000000001

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.4

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

   7. Simplified 85.4%

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

   8. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 51.9

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

   10. Simplified 51.9%

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

   11. Taylor expanded in kx around 0

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

       1. /-lowering-/.f64 N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. *-lowering-*.f64 N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. sin-lowering-sin.f64 N/A

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

       11. sin-lowering-sin.f64 52.1

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

   13. Simplified 52.1%

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

   if 1.1200000000000001 < kx

   1. Initial program 99.4%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right)} \cdot \left(2 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right)\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}\right) + 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) - \frac{2}{3}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}\right) + \color{blue}{\frac{-2}{3}}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right)}, 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{4}{45} + \frac{-2}{315} \cdot {ky}^{2}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-2}{315} \cdot {ky}^{2} + \frac{4}{45}}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \frac{-2}{315}} + \frac{4}{45}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-2}{315}, \frac{4}{45}\right)}, \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-2}{315}, \frac{4}{45}\right), \frac{-2}{3}\right), 2\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

      17. *-lowering-*.f64 50.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   7. Simplified 50.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, -0.006349206349206349, 0.08888888888888889\right), -0.6666666666666666\right), 2\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}} \]

   8. Taylor expanded in ky around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. cos-neg N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 51.4

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

   10. Simplified 51.4%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.25 \cdot 10^{-45}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.12:\\ \;\;\;\;\frac{\left(\sin ky \cdot \sin th\right) \cdot \mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(\cos \left(kx \cdot -2\right), -0.5, 0.5\right)}}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.12:\\ \;\;\;\;\frac{\left(\sin ky \cdot \sin th\right) \cdot \mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2e-45)
   (sin th)
   (if (<= kx 1.12)
     (/ (* (* (sin ky) (sin th)) (fma 0.16666666666666666 (* kx kx) 1.0)) kx)
     (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2e-45) {
		tmp = sin(th);
	} else if (kx <= 1.12) {
		tmp = ((sin(ky) * sin(th)) * fma(0.16666666666666666, (kx * kx), 1.0)) / kx;
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2e-45)
		tmp = sin(th);
	elseif (kx <= 1.12)
		tmp = Float64(Float64(Float64(sin(ky) * sin(th)) * fma(0.16666666666666666, Float64(kx * kx), 1.0)) / kx);
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2e-45], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.12], N[(N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.99999999999999997e-45

   1. Initial program 90.2%

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

   3. Taylor expanded in kx around 0

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

      1. sin-lowering-sin.f64 26.3

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

   5. Simplified 26.3%

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

   if 1.99999999999999997e-45 < kx < 1.1200000000000001

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-hypot.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. sin-lowering-sin.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.4

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

   7. Simplified 85.4%

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

   8. Taylor expanded in ky around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 51.9

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

   10. Simplified 51.9%

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

   11. Taylor expanded in kx around 0

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

       1. /-lowering-/.f64 N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft1-in N/A

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

       4. *-lowering-*.f64 N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, kx \cdot kx, 1\right) \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)}}{kx} \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, kx \cdot kx, 1\right) \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right)}{kx} \]

       11. sin-lowering-sin.f64 52.1

           \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right) \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right)}{kx} \]

   13. Simplified 52.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right) \cdot \left(\sin th \cdot \sin ky\right)}{kx}} \]

   if 1.1200000000000001 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}{\sin ky}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      6. +-commutative N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \]

      7. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. *-lowering-*.f64 51.4

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 51.4%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.12:\\ \;\;\;\;\frac{\left(\sin ky \cdot \sin th\right) \cdot \mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-184}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 5e-184)
   (* (sin th) (/ ky (* kx (fma -0.16666666666666666 (* kx kx) 1.0))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 5e-184) {
		tmp = sin(th) * (ky / (kx * fma(-0.16666666666666666, (kx * kx), 1.0)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 5e-184)
		tmp = Float64(sin(th) * Float64(ky / Float64(kx * fma(-0.16666666666666666, Float64(kx * kx), 1.0))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-184], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 5.00000000000000003e-184

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. sin-lowering-sin.f64 99.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied egg-rr 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)}\right)} \cdot \sin th \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)}\right)} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right)\right)} \cdot \sin th \]

      5. *-lowering-*.f64 60.4

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right)\right)} \cdot \sin th \]

   7. Simplified 60.4%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)}\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}} \cdot \sin th \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{ky}{kx \cdot \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)}} \cdot \sin th \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{ky}{kx \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{ky}{kx \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right)} \cdot \sin th \]

      6. *-lowering-*.f64 18.0

         \[\leadsto \frac{ky}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right)} \cdot \sin th \]

   10. Simplified 18.0%

       \[\leadsto \color{blue}{\frac{ky}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)}} \cdot \sin th \]

   if 5.00000000000000003e-184 < (sin.f64 ky)

   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 kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 55.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 55.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-184}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 32.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.05 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.05e-183) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.05e-183) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.05d-183) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.05e-183) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.05e-183:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.05e-183)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.05e-183)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.05e-183], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.0500000000000001e-183

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. sin-lowering-sin.f64 28.1

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 28.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.0500000000000001e-183 < ky

   1. Initial program 97.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 32.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 32.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.05 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 21.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 8 \cdot 10^{-213}:\\ \;\;\;\;2 \cdot \frac{\sin th \cdot \left(ky \cdot ky\right)}{kx \cdot kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 8e-213) (* 2.0 (/ (* (sin th) (* ky ky)) (* kx kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8e-213) {
		tmp = 2.0 * ((sin(th) * (ky * ky)) / (kx * kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 8d-213) then
        tmp = 2.0d0 * ((sin(th) * (ky * ky)) / (kx * kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8e-213) {
		tmp = 2.0 * ((Math.sin(th) * (ky * ky)) / (kx * kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 8e-213:
		tmp = 2.0 * ((math.sin(th) * (ky * ky)) / (kx * kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 8e-213)
		tmp = Float64(2.0 * Float64(Float64(sin(th) * Float64(ky * ky)) / Float64(kx * kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 8e-213)
		tmp = 2.0 * ((sin(th) * (ky * ky)) / (kx * kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 8e-213], N[(2.0 * N[(N[(N[Sin[th], $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] / N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 7.9999999999999996e-213

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. sin-lowering-sin.f64 22.1

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Simplified 22.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{ky}^{2} \cdot \sin th}{{kx}^{2}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{{ky}^{2} \cdot \sin th}{{kx}^{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{ky}^{2} \cdot \sin th}{{kx}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\color{blue}{\sin th \cdot {ky}^{2}}}{{kx}^{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\color{blue}{\sin th \cdot {ky}^{2}}}{{kx}^{2}} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto 2 \cdot \frac{\color{blue}{\sin th} \cdot {ky}^{2}}{{kx}^{2}} \]

      6. unpow2 N/A

         \[\leadsto 2 \cdot \frac{\sin th \cdot \color{blue}{\left(ky \cdot ky\right)}}{{kx}^{2}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\sin th \cdot \color{blue}{\left(ky \cdot ky\right)}}{{kx}^{2}} \]

      8. unpow2 N/A

         \[\leadsto 2 \cdot \frac{\sin th \cdot \left(ky \cdot ky\right)}{\color{blue}{kx \cdot kx}} \]

      9. *-lowering-*.f64 10.5

         \[\leadsto 2 \cdot \frac{\sin th \cdot \left(ky \cdot ky\right)}{\color{blue}{kx \cdot kx}} \]

   8. Simplified 10.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\sin th \cdot \left(ky \cdot ky\right)}{kx \cdot kx}} \]

   if 7.9999999999999996e-213 < ky

   1. Initial program 95.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 30.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 21.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7 \cdot 10^{-213}:\\ \;\;\;\;th \cdot \left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 7e-213) (* th (* -0.16666666666666666 (* th th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7e-213) {
		tmp = th * (-0.16666666666666666 * (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 (ky <= 7d-213) then
        tmp = th * ((-0.16666666666666666d0) * (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 (ky <= 7e-213) {
		tmp = th * (-0.16666666666666666 * (th * th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 7e-213:
		tmp = th * (-0.16666666666666666 * (th * th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 7e-213)
		tmp = Float64(th * Float64(-0.16666666666666666 * Float64(th * th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 7e-213)
		tmp = th * (-0.16666666666666666 * (th * th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 7e-213], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 7.00000000000000034e-213

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 15.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 15.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. *-lowering-*.f64 10.0

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 10.0%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)} \]

       2. unpow2 N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       3. *-lowering-*.f64 11.4

          \[\leadsto th \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 11.4%

       \[\leadsto th \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)} \]

   if 7.00000000000000034e-213 < ky

   1. Initial program 95.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 30.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 15.6% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.05 \cdot 10^{-183}:\\ \;\;\;\;th \cdot \left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.05e-183) (* th (* -0.16666666666666666 (* th th))) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.05e-183) {
		tmp = th * (-0.16666666666666666 * (th * th));
	} else {
		tmp = 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 (ky <= 1.05d-183) then
        tmp = th * ((-0.16666666666666666d0) * (th * th))
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.05e-183) {
		tmp = th * (-0.16666666666666666 * (th * th));
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.05e-183:
		tmp = th * (-0.16666666666666666 * (th * th))
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.05e-183)
		tmp = Float64(th * Float64(-0.16666666666666666 * Float64(th * th)));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.05e-183)
		tmp = th * (-0.16666666666666666 * (th * th));
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.05e-183], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 1.0500000000000001e-183

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 15.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 15.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. *-lowering-*.f64 9.7

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 9.7%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right)} \]

       2. unpow2 N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       3. *-lowering-*.f64 12.1

          \[\leadsto th \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 12.1%

       \[\leadsto th \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(th \cdot th\right)\right)} \]

   if 1.0500000000000001e-183 < ky

   1. Initial program 97.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 32.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 32.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th} \]
   7. Step-by-step derivation

   8. Simplified 16.3%

      \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 15.6% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-184}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.2e-184) (* -0.16666666666666666 (* th (* th th))) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-184) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = 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 (ky <= 4.2d-184) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-184) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.2e-184:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.2e-184)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.2e-184)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.2e-184], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 4.1999999999999998e-184

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 15.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 15.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. *-lowering-*.f64 9.7

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 9.7%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. *-lowering-*.f64 12.1

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 12.1%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 4.1999999999999998e-184 < ky

   1. Initial program 97.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 32.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 32.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th} \]
   7. Step-by-step derivation

   8. Simplified 16.3%

      \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 13.5% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 21.2

      \[\leadsto \color{blue}{\sin th} \]

5. Simplified 21.2%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th} \]
7. Step-by-step derivation

8. Simplified 12.0%

   \[\leadsto \color{blue}{th} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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)))