Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 14.2s

Alternatives: 27

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

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 94.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 78.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \cos \left(ky + ky\right)\\ t_3 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right), \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{kx \cdot kx + \mathsf{fma}\left(t\_2, -0.5, 0.5\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq -0.18:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot t\_2\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 0.35:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.999998:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (cos (+ ky ky)))
        (t_3
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* -0.16666666666666666 (* ky ky)) ky) (sin kx))))))
   (if (<= t_1 -1.0)
     (/ (sin th) (/ (sqrt (+ (* kx kx) (fma t_2 -0.5 0.5))) (sin ky)))
     (if (<= t_1 -0.18)
       (*
        (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 t_2)))))
        (* (sin ky) th))
       (if (<= t_1 0.35)
         t_3
         (if (<= t_1 0.999998)
           (*
            th
            (*
             (sqrt
              (/
               1.0
               (fma
                0.5
                (- 1.0 (cos (* kx -2.0)))
                (fma -0.5 (cos (* ky -2.0)) 0.5))))
             (sin ky)))
           (if (<= t_1 1.0) (sin th) t_3)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = cos((ky + ky));
	double t_3 = sin(th) * (sin(ky) / hypot(fma(ky, (-0.16666666666666666 * (ky * ky)), ky), sin(kx)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = sin(th) / (sqrt(((kx * kx) + fma(t_2, -0.5, 0.5))) / sin(ky));
	} else if (t_1 <= -0.18) {
		tmp = sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * t_2))))) * (sin(ky) * th);
	} else if (t_1 <= 0.35) {
		tmp = t_3;
	} else if (t_1 <= 0.999998) {
		tmp = th * (sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), fma(-0.5, cos((ky * -2.0)), 0.5)))) * sin(ky));
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = cos(Float64(ky + ky))
	t_3 = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky), sin(kx))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(sin(th) / Float64(sqrt(Float64(Float64(kx * kx) + fma(t_2, -0.5, 0.5))) / sin(ky)));
	elseif (t_1 <= -0.18)
		tmp = Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * t_2))))) * Float64(sin(ky) * th));
	elseif (t_1 <= 0.35)
		tmp = t_3;
	elseif (t_1 <= 0.999998)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))) * sin(ky)));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(t$95$2 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.18], N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.35], t$95$3, If[LessEqual[t$95$1, 0.999998], N[(th * N[(N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 64.7%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.17999999999999999

   1. Initial program 99.1%

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.3

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

   7. Applied rewrites 47.3%

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

   if -0.17999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.34999999999999998 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 0.34999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999998000000000054

   1. Initial program 99.3%

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 50.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 50.2%

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

   if 0.999998000000000054 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{kx \cdot kx + \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.18:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.35:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.999998:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right), \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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)))