Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.6%

Time: 11.3s

Alternatives: 25

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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-hypot.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. associate-/r/ N/A

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

   8. times-frac N/A

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

   9. clear-num N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow-1 N/A

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

   12. remove-double-div N/A

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

   13. clear-num N/A

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

   14. un-div-inv N/A

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

   15. lower-/.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 2: 83.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.04:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.8:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{{ky}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx))))))
   (if (<= t_3 -0.999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 -0.04)
       t_4
       (if (<= t_3 0.05)
         (* (/ (sin ky) (sqrt (+ t_2 (* ky ky)))) (sin th))
         (if (<= t_3 0.8)
           t_4
           (if (<= t_3 2.0)
             (sin th)
             (/
              (* (sin th) (/ 1.0 (hypot (sin kx) (sin ky))))
              (pow ky -1.0)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= -0.04) {
		tmp = t_4;
	} else if (t_3 <= 0.05) {
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	} else if (t_3 <= 0.8) {
		tmp = t_4;
	} else if (t_3 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) * (1.0 / hypot(sin(kx), sin(ky)))) / pow(ky, -1.0);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + t_1));
	double t_4 = (-th * Math.sin(ky)) * (-1.0 / Math.hypot(Math.sin(ky), Math.sin(kx)));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_3 <= -0.04) {
		tmp = t_4;
	} else if (t_3 <= 0.05) {
		tmp = (Math.sin(ky) / Math.sqrt((t_2 + (ky * ky)))) * Math.sin(th);
	} else if (t_3 <= 0.8) {
		tmp = t_4;
	} else if (t_3 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) * (1.0 / Math.hypot(Math.sin(kx), Math.sin(ky)))) / Math.pow(ky, -1.0);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + t_1))
	t_4 = (-th * math.sin(ky)) * (-1.0 / math.hypot(math.sin(ky), math.sin(kx)))
	tmp = 0
	if t_3 <= -0.999:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_3 <= -0.04:
		tmp = t_4
	elif t_3 <= 0.05:
		tmp = (math.sin(ky) / math.sqrt((t_2 + (ky * ky)))) * math.sin(th)
	elif t_3 <= 0.8:
		tmp = t_4
	elif t_3 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) * (1.0 / math.hypot(math.sin(kx), math.sin(ky)))) / math.pow(ky, -1.0)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= -0.04)
		tmp = t_4;
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))) * sin(th));
	elseif (t_3 <= 0.8)
		tmp = t_4;
	elseif (t_3 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) * Float64(1.0 / hypot(sin(kx), sin(ky)))) / (ky ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + t_1));
	t_4 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	tmp = 0.0;
	if (t_3 <= -0.999)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_3 <= -0.04)
		tmp = t_4;
	elseif (t_3 <= 0.05)
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	elseif (t_3 <= 0.8)
		tmp = t_4;
	elseif (t_3 <= 2.0)
		tmp = sin(th);
	else
		tmp = (sin(th) * (1.0 / hypot(sin(kx), sin(ky)))) / (ky ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.04], t$95$4, If[LessEqual[t$95$3, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.8], t$95$4, If[LessEqual[t$95$3, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[ky, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 83.1

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

   5. Applied rewrites 83.1%

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

   if -0.998999999999999999 < (/.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))))) < -0.0400000000000000008 or 0.050000000000000003 < (/.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))))) < 0.80000000000000004

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 47.4

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

   7. Applied rewrites 47.4%

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

   if -0.0400000000000000008 < (/.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))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 0.80000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 86.9

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

   5. Applied rewrites 86.9%

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

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

   1. Initial program 2.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. distribute-neg-frac2 N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.3

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

      14. lift-sin.f64 N/A

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

      15. lift-hypot.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-hypot.f64 N/A

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

      18. lift-sin.f64 99.3

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

   9. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\left(-\sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}}{\frac{1}{ky}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.8:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{{ky}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + t\_1}}\\ t_5 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{t\_2}\\ \mathbf{if}\;t\_4 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.04:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.8:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{t\_2}}{{ky}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (hypot (sin ky) (sin kx)))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 t_1))))
        (t_5 (* (* (- th) (sin ky)) (/ -1.0 t_2))))
   (if (<= t_4 -0.999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_4 -0.04)
       t_5
       (if (<= t_4 0.05)
         (* (/ (sin ky) (sqrt (+ t_3 (* ky ky)))) (sin th))
         (if (<= t_4 0.8)
           t_5
           (if (<= t_4 2.0) (sin th) (/ (/ (sin th) t_2) (pow ky -1.0)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + t_1));
	double t_5 = (-th * sin(ky)) * (-1.0 / t_2);
	double tmp;
	if (t_4 <= -0.999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_4 <= -0.04) {
		tmp = t_5;
	} else if (t_4 <= 0.05) {
		tmp = (sin(ky) / sqrt((t_3 + (ky * ky)))) * sin(th);
	} else if (t_4 <= 0.8) {
		tmp = t_5;
	} else if (t_4 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / t_2) / pow(ky, -1.0);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_3 = Math.pow(Math.sin(kx), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_3 + t_1));
	double t_5 = (-th * Math.sin(ky)) * (-1.0 / t_2);
	double tmp;
	if (t_4 <= -0.999) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_4 <= -0.04) {
		tmp = t_5;
	} else if (t_4 <= 0.05) {
		tmp = (Math.sin(ky) / Math.sqrt((t_3 + (ky * ky)))) * Math.sin(th);
	} else if (t_4 <= 0.8) {
		tmp = t_5;
	} else if (t_4 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) / t_2) / Math.pow(ky, -1.0);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.hypot(math.sin(ky), math.sin(kx))
	t_3 = math.pow(math.sin(kx), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_3 + t_1))
	t_5 = (-th * math.sin(ky)) * (-1.0 / t_2)
	tmp = 0
	if t_4 <= -0.999:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_4 <= -0.04:
		tmp = t_5
	elif t_4 <= 0.05:
		tmp = (math.sin(ky) / math.sqrt((t_3 + (ky * ky)))) * math.sin(th)
	elif t_4 <= 0.8:
		tmp = t_5
	elif t_4 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) / t_2) / math.pow(ky, -1.0)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = hypot(sin(ky), sin(kx))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + t_1)))
	t_5 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / t_2))
	tmp = 0.0
	if (t_4 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_4 <= -0.04)
		tmp = t_5;
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_3 + Float64(ky * ky)))) * sin(th));
	elseif (t_4 <= 0.8)
		tmp = t_5;
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / t_2) / (ky ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = hypot(sin(ky), sin(kx));
	t_3 = sin(kx) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_3 + t_1));
	t_5 = (-th * sin(ky)) * (-1.0 / t_2);
	tmp = 0.0;
	if (t_4 <= -0.999)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_4 <= -0.04)
		tmp = t_5;
	elseif (t_4 <= 0.05)
		tmp = (sin(ky) / sqrt((t_3 + (ky * ky)))) * sin(th);
	elseif (t_4 <= 0.8)
		tmp = t_5;
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = (sin(th) / t_2) / (ky ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.04], t$95$5, If[LessEqual[t$95$4, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.8], t$95$5, If[LessEqual[t$95$4, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / t$95$2), $MachinePrecision] / N[Power[ky, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 83.1

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

   5. Applied rewrites 83.1%

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

   if -0.998999999999999999 < (/.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))))) < -0.0400000000000000008 or 0.050000000000000003 < (/.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))))) < 0.80000000000000004

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 47.4

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

   7. Applied rewrites 47.4%

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

   if -0.0400000000000000008 < (/.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))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 0.80000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 86.9

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

   5. Applied rewrites 86.9%

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

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

   1. Initial program 2.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\color{blue}{\frac{1}{ky}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.8:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{{ky}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ t_3 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_4 := \frac{\frac{\sin th}{t\_3}}{{ky}^{-1}}\\ t_5 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{t\_3}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.8:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1))))
        (t_3 (hypot (sin ky) (sin kx)))
        (t_4 (/ (/ (sin th) t_3) (pow ky -1.0)))
        (t_5 (* (* (- th) (sin ky)) (/ -1.0 t_3))))
   (if (<= t_2 -0.999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.04)
       t_5
       (if (<= t_2 0.05)
         t_4
         (if (<= t_2 0.8) t_5 (if (<= t_2 2.0) (sin th) t_4)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double t_3 = hypot(sin(ky), sin(kx));
	double t_4 = (sin(th) / t_3) / pow(ky, -1.0);
	double t_5 = (-th * sin(ky)) * (-1.0 / t_3);
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.04) {
		tmp = t_5;
	} else if (t_2 <= 0.05) {
		tmp = t_4;
	} else if (t_2 <= 0.8) {
		tmp = t_5;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double t_3 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_4 = (Math.sin(th) / t_3) / Math.pow(ky, -1.0);
	double t_5 = (-th * Math.sin(ky)) * (-1.0 / t_3);
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_2 <= -0.04) {
		tmp = t_5;
	} else if (t_2 <= 0.05) {
		tmp = t_4;
	} else if (t_2 <= 0.8) {
		tmp = t_5;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	t_3 = math.hypot(math.sin(ky), math.sin(kx))
	t_4 = (math.sin(th) / t_3) / math.pow(ky, -1.0)
	t_5 = (-th * math.sin(ky)) * (-1.0 / t_3)
	tmp = 0
	if t_2 <= -0.999:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_2 <= -0.04:
		tmp = t_5
	elif t_2 <= 0.05:
		tmp = t_4
	elif t_2 <= 0.8:
		tmp = t_5
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = t_4
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	t_3 = hypot(sin(ky), sin(kx))
	t_4 = Float64(Float64(sin(th) / t_3) / (ky ^ -1.0))
	t_5 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / t_3))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.04)
		tmp = t_5;
	elseif (t_2 <= 0.05)
		tmp = t_4;
	elseif (t_2 <= 0.8)
		tmp = t_5;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	t_3 = hypot(sin(ky), sin(kx));
	t_4 = (sin(th) / t_3) / (ky ^ -1.0);
	t_5 = (-th * sin(ky)) * (-1.0 / t_3);
	tmp = 0.0;
	if (t_2 <= -0.999)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_2 <= -0.04)
		tmp = t_5;
	elseif (t_2 <= 0.05)
		tmp = t_4;
	elseif (t_2 <= 0.8)
		tmp = t_5;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] / N[Power[ky, -1.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.04], t$95$5, If[LessEqual[t$95$2, 0.05], t$95$4, If[LessEqual[t$95$2, 0.8], t$95$5, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$4]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 83.1

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

   5. Applied rewrites 83.1%

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

   if -0.998999999999999999 < (/.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))))) < -0.0400000000000000008 or 0.050000000000000003 < (/.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))))) < 0.80000000000000004

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.4

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 47.4

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

   7. Applied rewrites 47.4%

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

   if -0.0400000000000000008 < (/.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))))) < 0.050000000000000003 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 0.80000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{{ky}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.8:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{{ky}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-\sin th\right) \cdot \frac{-1}{\sin kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/ (* (- (sin th)) (/ -1.0 (sin kx))) (pow ky -1.0))
   (sin th)))

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = (-sin(th) * ((-1.0d0) / sin(kx))) / (ky ** (-1.0d0))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = (-Math.sin(th) * (-1.0 / Math.sin(kx))) / Math.pow(ky, -1.0);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-5:
		tmp = (-math.sin(th) * (-1.0 / math.sin(kx))) / math.pow(ky, -1.0)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(Float64(-sin(th)) * Float64(-1.0 / sin(kx))) / (ky ^ -1.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = (-sin(th) * (-1.0 / sin(kx))) / (ky ^ -1.0);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. distribute-neg-frac2 N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 66.8

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

      14. lift-sin.f64 N/A

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

      15. lift-hypot.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-hypot.f64 N/A

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

      18. lift-sin.f64 66.8

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in ky around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 37.4

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

   12. Applied rewrites 37.4%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

4. Final simplification 43.4%

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

Alternative 6: 44.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/ (/ (sin th) (sin kx)) (pow ky -1.0))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (sin(th) / sin(kx)) / pow(ky, -1.0);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = (sin(th) / sin(kx)) / (ky ** (-1.0d0))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = (Math.sin(th) / Math.sin(kx)) / Math.pow(ky, -1.0);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-5:
		tmp = (math.sin(th) / math.sin(kx)) / math.pow(ky, -1.0)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(sin(th) / sin(kx)) / (ky ^ -1.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = (sin(th) / sin(kx)) / (ky ^ -1.0);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 37.4

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

   10. Applied rewrites 37.4%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/ (sin ky) (/ (sin kx) (sin th)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-5:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = sin(ky) / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow-1 N/A

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

      12. remove-double-div N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 38.3

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

   9. Applied rewrites 38.3%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 8: 45.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (* (/ (sin ky) (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 9: 44.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, ky \cdot ky, 0.019444444444444445\right), ky \cdot ky, 0.16666666666666666\right), ky \cdot ky, 1\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/
    (/ (sin th) (sin kx))
    (/
     (fma
      (fma
       (fma 0.00205026455026455 (* ky ky) 0.019444444444444445)
       (* ky ky)
       0.16666666666666666)
      (* ky ky)
      1.0)
     ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (sin(th) / sin(kx)) / (fma(fma(fma(0.00205026455026455, (ky * ky), 0.019444444444444445), (ky * ky), 0.16666666666666666), (ky * ky), 1.0) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(sin(th) / sin(kx)) / Float64(fma(fma(fma(0.00205026455026455, Float64(ky * ky), 0.019444444444444445), Float64(ky * ky), 0.16666666666666666), Float64(ky * ky), 1.0) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.00205026455026455 * N[(ky * ky), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 37.4

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{1 + {ky}^{2} \cdot \left(\frac{1}{6} + {ky}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right)\right)}{ky}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{1 + {ky}^{2} \cdot \left(\frac{1}{6} + {ky}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right)\right)}{ky}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{{ky}^{2} \cdot \left(\frac{1}{6} + {ky}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right)\right) + 1}}{ky}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{\left(\frac{1}{6} + {ky}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right)\right) \cdot {ky}^{2}} + 1}{ky}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {ky}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right), {ky}^{2}, 1\right)}}{ky}} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\color{blue}{{ky}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right) + \frac{1}{6}}, {ky}^{2}, 1\right)}{ky}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}\right) \cdot {ky}^{2}} + \frac{1}{6}, {ky}^{2}, 1\right)}{ky}} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360} + \frac{31}{15120} \cdot {ky}^{2}, {ky}^{2}, \frac{1}{6}\right)}, {ky}^{2}, 1\right)}{ky}} \]

       8. +-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{31}{15120} \cdot {ky}^{2} + \frac{7}{360}}, {ky}^{2}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{31}{15120}, {ky}^{2}, \frac{7}{360}\right)}, {ky}^{2}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       10. unpow2 N/A

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, \color{blue}{ky \cdot ky}, \frac{7}{360}\right), {ky}^{2}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, \color{blue}{ky \cdot ky}, \frac{7}{360}\right), {ky}^{2}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       12. unpow2 N/A

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, ky \cdot ky, \frac{7}{360}\right), \color{blue}{ky \cdot ky}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, ky \cdot ky, \frac{7}{360}\right), \color{blue}{ky \cdot ky}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       14. unpow2 N/A

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{15120}, ky \cdot ky, \frac{7}{360}\right), ky \cdot ky, \frac{1}{6}\right), \color{blue}{ky \cdot ky}, 1\right)}{ky}} \]

       15. lower-*.f64 37.8

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, ky \cdot ky, 0.019444444444444445\right), ky \cdot ky, 0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right)}{ky}} \]

   13. Applied rewrites 37.8%

       \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, ky \cdot ky, 0.019444444444444445\right), ky \cdot ky, 0.16666666666666666\right), ky \cdot ky, 1\right)}{ky}}} \]

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 10: 44.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, ky \cdot ky, 0.16666666666666666\right), ky \cdot ky, 1\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/
    (/ (sin th) (sin kx))
    (/
     (fma
      (fma 0.019444444444444445 (* ky ky) 0.16666666666666666)
      (* ky ky)
      1.0)
     ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (sin(th) / sin(kx)) / (fma(fma(0.019444444444444445, (ky * ky), 0.16666666666666666), (ky * ky), 1.0) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(sin(th) / sin(kx)) / Float64(fma(fma(0.019444444444444445, Float64(ky * ky), 0.16666666666666666), Float64(ky * ky), 1.0) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.019444444444444445 * N[(ky * ky), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 37.4

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{1 + {ky}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {ky}^{2}\right)}{ky}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{1 + {ky}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {ky}^{2}\right)}{ky}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{{ky}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {ky}^{2}\right) + 1}}{ky}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {ky}^{2}\right) \cdot {ky}^{2}} + 1}{ky}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {ky}^{2}, {ky}^{2}, 1\right)}}{ky}} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {ky}^{2} + \frac{1}{6}}, {ky}^{2}, 1\right)}{ky}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {ky}^{2}, \frac{1}{6}\right)}, {ky}^{2}, 1\right)}{ky}} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{ky \cdot ky}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{ky \cdot ky}, \frac{1}{6}\right), {ky}^{2}, 1\right)}{ky}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, ky \cdot ky, \frac{1}{6}\right), \color{blue}{ky \cdot ky}, 1\right)}{ky}} \]

       10. lower-*.f64 37.9

           \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, ky \cdot ky, 0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right)}{ky}} \]

   13. Applied rewrites 37.9%

       \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, ky \cdot ky, 0.16666666666666666\right), ky \cdot ky, 1\right)}{ky}}} \]

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 11: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{\mathsf{fma}\left(ky \cdot ky, 0.16666666666666666, 1\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/ (/ (sin th) (sin kx)) (/ (fma (* ky ky) 0.16666666666666666 1.0) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (sin(th) / sin(kx)) / (fma((ky * ky), 0.16666666666666666, 1.0) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(sin(th) / sin(kx)) / Float64(fma(Float64(ky * ky), 0.16666666666666666, 1.0) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 37.4

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in ky around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot {ky}^{2}}{ky}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{\frac{1}{6} \cdot {ky}^{2} + 1}}{ky}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{\sin th}{\sin kx}}{\frac{\color{blue}{{ky}^{2} \cdot \frac{1}{6}} + 1}{ky}} \]

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 37.8

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

   13. Applied rewrites 37.8%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 12: 35.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\sin th}{kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-27)
   (/ (/ (sin th) kx) (pow ky -1.0))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-27) {
		tmp = (sin(th) / kx) / pow(ky, -1.0);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-27) then
        tmp = (sin(th) / kx) / (ky ** (-1.0d0))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-27) {
		tmp = (Math.sin(th) / kx) / Math.pow(ky, -1.0);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-27:
		tmp = (math.sin(th) / kx) / math.pow(ky, -1.0)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-27)
		tmp = Float64(Float64(sin(th) / kx) / (ky ^ -1.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-27)
		tmp = (sin(th) / kx) / (ky ^ -1.0);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-27], N[(N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision] / N[Power[ky, -1.0], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.0000000000000002e-27

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 37.3

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

   10. Applied rewrites 37.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 21.2%

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

   if 4.0000000000000002e-27 < (/.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 91.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 54.4

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

   5. Applied rewrites 54.4%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\sin th}{kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{th}{\sin kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/ (/ th (sin kx)) (pow ky -1.0))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (th / sin(kx)) / pow(ky, -1.0);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = (th / sin(kx)) / (ky ** (-1.0d0))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = (th / Math.sin(kx)) / Math.pow(ky, -1.0);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-5:
		tmp = (th / math.sin(kx)) / math.pow(ky, -1.0)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(th / sin(kx)) / (ky ^ -1.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = (th / sin(kx)) / (ky ^ -1.0);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 66.8

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 37.4

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in th around 0

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

   13. Applied rewrites 23.5%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{th}{\sin kx}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-5:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 37.5

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

   7. Applied rewrites 37.5%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 15: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-5)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 95.2%

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

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

      2. lower-sin.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if 4.00000000000000033e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 16: 30.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-79}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-79)
   (* (pow th 3.0) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-79) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1d-79) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1e-79) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1e-79:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-79)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-79)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-79], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-79

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 13.6%

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

   if 1e-79 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 48.0

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

   5. Applied rewrites 48.0%

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

Alternative 17: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 7.1e-6)
   (/ (* (sin th) (/ 1.0 (hypot (sin kx) (sin ky)))) (pow ky -1.0))
   (/
    (* (sin th) (sin ky))
    (/
     (sqrt (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
     2.0))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7.1e-6) {
		tmp = (sin(th) * (1.0 / hypot(sin(kx), sin(ky)))) / pow(ky, -1.0);
	} else {
		tmp = (sin(th) * sin(ky)) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 7.1e-6)
		tmp = Float64(Float64(sin(th) * Float64(1.0 / hypot(sin(kx), sin(ky)))) / (ky ^ -1.0));
	else
		tmp = Float64(Float64(sin(th) * sin(ky)) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 7.1e-6], N[(N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[ky, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 7.0999999999999998e-6

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 71.7

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

   7. Applied rewrites 71.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. distribute-neg-frac2 N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 71.7

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

      14. lift-sin.f64 N/A

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

      15. lift-hypot.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-hypot.f64 N/A

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

      18. lift-sin.f64 71.7

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

   9. Applied rewrites 71.7%

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

   if 7.0999999999999998e-6 < ky

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. sin-mult N/A

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

      7. frac-add N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

   6. Applied rewrites 97.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 7.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 7.1e-6)
   (/ (* (sin th) (/ 1.0 (hypot (sin kx) (sin ky)))) (pow ky -1.0))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* ky 2.0))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7.1e-6) {
		tmp = (sin(th) * (1.0 / hypot(sin(kx), sin(ky)))) / pow(ky, -1.0);
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 7.1e-6)
		tmp = Float64(Float64(sin(th) * Float64(1.0 / hypot(sin(kx), sin(ky)))) / (ky ^ -1.0));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 7.1e-6], N[(N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[ky, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 7.0999999999999998e-6

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 71.7

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

   7. Applied rewrites 71.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. distribute-neg-frac2 N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 71.7

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

      14. lift-sin.f64 N/A

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

      15. lift-hypot.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-hypot.f64 N/A

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

      18. lift-sin.f64 71.7

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

   9. Applied rewrites 71.7%

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

   if 7.0999999999999998e-6 < ky

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 97.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 7.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{{ky}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 93.6

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

   8. lift-sqrt.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-hypot.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 20: 63.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 5.9e-5)
   (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
   (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 5.9e-5) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 5.9e-5) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = (Math.sin(th) * ky) / Math.hypot(Math.sin(ky), Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 5.9e-5:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = (math.sin(th) * ky) / math.hypot(math.sin(ky), math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 5.9e-5)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 5.9e-5)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 5.9e-5], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 5.8999999999999998e-5

   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-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. times-frac N/A

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

      9. clear-num N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow-1 N/A

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

      12. remove-double-div N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

   if 5.8999999999999998e-5 < th

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 60.3

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

   7. Applied rewrites 60.3%

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

Alternative 21: 59.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= th 5.9e-5) (/ (* (sin ky) th) t_1) (/ (* (sin th) ky) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (th <= 5.9e-5) {
		tmp = (sin(ky) * th) / t_1;
	} else {
		tmp = (sin(th) * ky) / t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (th <= 5.9e-5) {
		tmp = (Math.sin(ky) * th) / t_1;
	} else {
		tmp = (Math.sin(th) * ky) / t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if th <= 5.9e-5:
		tmp = (math.sin(ky) * th) / t_1
	else:
		tmp = (math.sin(th) * ky) / t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (th <= 5.9e-5)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	else
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (th <= 5.9e-5)
		tmp = (sin(ky) * th) / t_1;
	else
		tmp = (sin(th) * ky) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, 5.9e-5], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 5.8999999999999998e-5

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.2

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

   7. Applied rewrites 58.2%

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

   if 5.8999999999999998e-5 < th

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 60.3

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

   7. Applied rewrites 60.3%

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

Alternative 22: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.5:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + ky \cdot ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 9.5)
   (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))
   (*
    (/ (sin ky) (sqrt (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (* ky ky))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 9.5) {
		tmp = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 9.5) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 9.5:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	else:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 9.5)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(ky * ky)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 9.5)
		tmp = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	else
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 9.5], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.5

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 57.7

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

   7. Applied rewrites 57.7%

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

   if 9.5 < th

   1. Initial program 93.2%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2 N/A

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

      12. lower-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

Alternative 23: 37.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.5 \cdot 10^{-173}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + ky \cdot ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.5e-173)
   (sin th)
   (if (<= kx 2.4e-7)
     (* (/ (sin ky) (sqrt (+ (* kx kx) (* ky ky)))) (sin th))
     (*
      (/ (sin ky) (sqrt (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (* ky ky))))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.5e-173) {
		tmp = sin(th);
	} else if (kx <= 2.4e-7) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * 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 (kx <= 8.5d-173) then
        tmp = sin(th)
    else if (kx <= 2.4d-7) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th)
    else
        tmp = (sin(ky) / sqrt(((0.5d0 - (cos((2.0d0 * kx)) * 0.5d0)) + (ky * ky)))) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.5e-173) {
		tmp = Math.sin(th);
	} else if (kx <= 2.4e-7) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8.5e-173:
		tmp = math.sin(th)
	elif kx <= 2.4e-7:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (ky * ky)))) * math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.5e-173)
		tmp = sin(th);
	elseif (kx <= 2.4e-7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(ky * ky)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8.5e-173)
		tmp = sin(th);
	elseif (kx <= 2.4e-7)
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	else
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (ky * ky)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.5e-173], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 2.4e-7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 8.4999999999999996e-173

   1. Initial program 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 23.9

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

   5. Applied rewrites 23.9%

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

   if 8.4999999999999996e-173 < kx < 2.39999999999999979e-7

   1. Initial program 92.3%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if 2.39999999999999979e-7 < kx

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 56.6

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 56.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + ky \cdot ky}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + ky \cdot ky}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + ky \cdot ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + ky \cdot ky}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + ky \cdot ky}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + ky \cdot ky}} \cdot \sin th \]

      7. count-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + ky \cdot ky}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + ky \cdot ky}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + ky \cdot ky}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      11. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      12. lower-*.f64 56.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + ky \cdot ky}} \cdot \sin th \]

   7. Applied rewrites 56.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + ky \cdot ky}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 23.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th);
}

Python

def code(kx, ky, th):
	return math.sin(th)

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 20.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 20.0%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 25: 12.6% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (fma (* -0.16666666666666666 (* th th)) th th))

C

double code(double kx, double ky, double th) {
	return fma((-0.16666666666666666 * (th * th)), th, th);
}

Julia

function code(kx, ky, th)
	return fma(Float64(-0.16666666666666666 * Float64(th * th)), th, th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] * th + th), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 20.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 20.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 11.0%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 11.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(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)))