Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 10.5s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)}{\sin ky}}\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 0.995)
       (* (/ (sin ky) (sqrt t_2)) (sin th))
       (/ (sin th) (/ (fma (* (/ 0.5 (sin ky)) kx) kx (sin ky)) (sin ky)))))))

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_1 + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else {
		tmp = sin(th) / (fma(((0.5 / sin(ky)) * kx), kx, sin(ky)) / sin(ky));
	}
	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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	else
		tmp = Float64(sin(th) / Float64(fma(Float64(Float64(0.5 / sin(ky)) * kx), kx, sin(ky)) / sin(ky)));
	end
	return 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], 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, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * kx), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 44.1

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

   7. Applied rewrites 44.1%

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

   9. Applied rewrites 73.1%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. *-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 87.8

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\frac{0.5}{\sin ky} \cdot kx, kx, \sin ky\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 0.995)
       (* (/ (sin ky) (sqrt t_2)) (sin th))
       (* (/ (sin ky) (fma (* 0.5 kx) (/ kx (sin ky)) (sin ky))) (sin th))))))

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_1 + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / sin(ky)), sin(ky))) * sin(th);
	}
	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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / sin(ky)), sin(ky))) * sin(th));
	end
	return 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], 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, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / N[Sin[ky], $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 44.1

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

   7. Applied rewrites 44.1%

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

   9. Applied rewrites 73.1%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.8%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)}{\sin ky}}\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 0.98)
       (* (/ (sin ky) (sqrt t_2)) (sin th))
       (/ (sin th) (/ (fma (* (/ kx ky) 0.5) kx (sin ky)) (sin ky)))))))

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_1 + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= 0.98) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else {
		tmp = sin(th) / (fma(((kx / ky) * 0.5), kx, sin(ky)) / sin(ky));
	}
	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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= 0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	else
		tmp = Float64(sin(th) / Float64(fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky)) / sin(ky)));
	end
	return 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], 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, 0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 44.1

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

   7. Applied rewrites 44.1%

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

   9. Applied rewrites 73.1%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. *-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 87.8

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 92.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.99:\\ \;\;\;\;\sqrt{\frac{1}{t\_1}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_3 \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)}{\sin ky}}\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -0.99)
     (* (sqrt (/ 1.0 t_1)) (* th (sin ky)))
     (if (<= t_3 0.98)
       (* (/ (sin ky) (sqrt t_2)) (sin th))
       (/ (sin th) (/ (fma (* (/ kx ky) 0.5) kx (sin ky)) (sin ky)))))))

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_1 + t_2));
	double tmp;
	if (t_3 <= -0.99) {
		tmp = sqrt((1.0 / t_1)) * (th * sin(ky));
	} else if (t_3 <= 0.98) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else {
		tmp = sin(th) / (fma(((kx / ky) * 0.5), kx, sin(ky)) / sin(ky));
	}
	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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.99)
		tmp = Float64(sqrt(Float64(1.0 / t_1)) * Float64(th * sin(ky)));
	elseif (t_3 <= 0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	else
		tmp = Float64(sin(th) / Float64(fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky)) / sin(ky)));
	end
	return 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.99], N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 92.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 37.5

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

   7. Applied rewrites 37.5%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 34.6%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 44.9

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

   7. Applied rewrites 44.9%

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

   9. Applied rewrites 74.9%

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

   if 0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. *-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 87.8

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 92.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.99:\\ \;\;\;\;\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.71:\\ \;\;\;\;\sqrt{\frac{1}{t\_1}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_2 \leq 0.03:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0))))))
   (if (<= t_2 -0.71)
     (* (sqrt (/ 1.0 t_1)) (* th (sin ky)))
     (if (<= t_2 0.03) (/ (sin th) (/ (sin kx) (sin ky))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.71) {
		tmp = sqrt((1.0 / t_1)) * (th * sin(ky));
	} else if (t_2 <= 0.03) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(ky) ** 2.0d0
    t_2 = sin(ky) / sqrt((t_1 + (sin(kx) ** 2.0d0)))
    if (t_2 <= (-0.71d0)) then
        tmp = sqrt((1.0d0 / t_1)) * (th * sin(ky))
    else if (t_2 <= 0.03d0) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.71) {
		tmp = Math.sqrt((1.0 / t_1)) * (th * Math.sin(ky));
	} else if (t_2 <= 0.03) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_2 <= -0.71:
		tmp = math.sqrt((1.0 / t_1)) * (th * math.sin(ky))
	elif t_2 <= 0.03:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.71)
		tmp = Float64(sqrt(Float64(1.0 / t_1)) * Float64(th * sin(ky)));
	elseif (t_2 <= 0.03)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt((t_1 + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.71)
		tmp = sqrt((1.0 / t_1)) * (th * sin(ky));
	elseif (t_2 <= 0.03)
		tmp = sin(th) / (sin(kx) / sin(ky));
	else
		tmp = sin(th);
	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[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.71], N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.03], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70999999999999996

   1. Initial program 93.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 37.7

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

   7. Applied rewrites 37.7%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 30.9%

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

   if -0.70999999999999996 < (/.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))))) < 0.029999999999999999

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-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 99.6

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

      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

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

      1. lower-sin.f64 55.7

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

   7. Applied rewrites 55.7%

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

   if 0.029999999999999999 < (/.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 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.71:\\ \;\;\;\;\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.03:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.03:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.03)
   (/ (sin th) (/ (sin kx) (sin ky)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.03) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.03d0) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.03) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.03:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.03)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.03)
		tmp = sin(th) / (sin(kx) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.03], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $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))))) < 0.029999999999999999

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.1

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

      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

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

      1. lower-sin.f64 36.7

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

   7. Applied rewrites 36.7%

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

   if 0.029999999999999999 < (/.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 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.03:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.03:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.03)
   (* (/ (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(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.03) {
		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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.03d0) 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.03) {
		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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.03:
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.03)
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.03)
		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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.03], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.029999999999999999

   1. Initial program 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   if 0.029999999999999999 < (/.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 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.03:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-6)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-6) {
		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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-6) 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-6) {
		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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-6:
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-6], 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))))) < 9.99999999999999955e-7

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.1

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

      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 35.4

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

   7. Applied rewrites 35.4%

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

   if 9.99999999999999955e-7 < (/.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 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-6)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-6) {
		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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-6) 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-6) {
		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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-6:
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-6)
		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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-6], 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))))) < 9.99999999999999955e-7

   1. Initial program 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 35.4

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

   5. Applied rewrites 35.4%

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

   if 9.99999999999999955e-7 < (/.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 91.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 5e-7)
     (/
      (sin th)
      (/
       (hypot (sin ky) (/ 1.0 (/ (fma 0.16666666666666666 (* kx kx) 1.0) kx)))
       (sin ky)))
     (* (/ (sin ky) (sqrt t_1)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 5e-7) {
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (fma(0.16666666666666666, (kx * kx), 1.0) / kx))) / sin(ky));
	} else {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 5e-7)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), Float64(1.0 / Float64(fma(0.16666666666666666, Float64(kx * kx), 1.0) / kx))) / sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(1.0 / N[(N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 4.99999999999999977e-7

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-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 91.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 4.99999999999999977e-7 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 43.4

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 68.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 2.15 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{kx}}\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 2.15e-12)
     (/ (sin th) (/ (hypot (sin ky) (/ 1.0 (/ 1.0 kx))) (sin ky)))
     (* (/ (sin ky) (sqrt t_1)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 2.15e-12) {
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (1.0 / kx))) / sin(ky));
	} else {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double tmp;
	if (t_1 <= 2.15e-12) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(ky), (1.0 / (1.0 / kx))) / Math.sin(ky));
	} else {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	tmp = 0
	if t_1 <= 2.15e-12:
		tmp = math.sin(th) / (math.hypot(math.sin(ky), (1.0 / (1.0 / kx))) / math.sin(ky))
	else:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 2.15e-12)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), Float64(1.0 / Float64(1.0 / kx))) / sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 2.15e-12)
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (1.0 / kx))) / sin(ky));
	else
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 2.15e-12], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(1.0 / N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.14999999999999993e-12

   1. Initial program 91.6%

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

      1. lift-*.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 91.7

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 2.14999999999999993e-12 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 42.9

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

   7. Applied rewrites 42.9%

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

   9. Applied rewrites 68.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 5e-15)
   (* (/ th (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 5e-15) {
		tmp = (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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 5d-15) then
        tmp = (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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 5e-15) {
		tmp = (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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 5e-15:
		tmp = (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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-15)
		tmp = Float64(Float64(th / 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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-15)
		tmp = (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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $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))))) < 4.99999999999999999e-15

   1. Initial program 97.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{\color{blue}{\left(\frac{1}{2} \cdot 4\right)}}}} \cdot \sin th \]

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. metadata-eval 26.0

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 41.4

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

   7. Applied rewrites 41.4%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 22.2%

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

   if 4.99999999999999999e-15 < (/.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 91.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 7.4 \cdot 10^{-99}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 7.4e-99)
   (* -0.16666666666666666 (pow th 3.0))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 7.4e-99) {
		tmp = -0.16666666666666666 * pow(th, 3.0);
	} 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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 7.4d-99) then
        tmp = (-0.16666666666666666d0) * (th ** 3.0d0)
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 7.4e-99) {
		tmp = -0.16666666666666666 * Math.pow(th, 3.0);
	} 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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 7.4e-99:
		tmp = -0.16666666666666666 * math.pow(th, 3.0)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 7.4e-99)
		tmp = Float64(-0.16666666666666666 * (th ^ 3.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 7.4e-99)
		tmp = -0.16666666666666666 * (th ^ 3.0);
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 7.4e-99], N[(-0.16666666666666666 * N[Power[th, 3.0], $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))))) < 7.400000000000001e-99

   1. Initial program 96.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 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 16.6%

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

   if 7.400000000000001e-99 < (/.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 92.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 7.4 \cdot 10^{-99}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.0023:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.0023)
   (/
    (sin th)
    (/
     (hypot (sin ky) (/ 1.0 (/ (fma 0.16666666666666666 (* kx kx) 1.0) kx)))
     (sin ky)))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* (- 1.0 (cos (* 2.0 kx))) 2.0)))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0023) {
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (fma(0.16666666666666666, (kx * kx), 1.0) / kx))) / sin(ky));
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, ((1.0 - cos((2.0 * kx))) * 2.0))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.0023)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), Float64(1.0 / Float64(fma(0.16666666666666666, Float64(kx * kx), 1.0) / kx))) / sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.0023], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(1.0 / N[(N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.0023

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.5

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

   9. Applied rewrites 79.5%

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

   if 0.0023 < kx

   1. Initial program 99.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.0023:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.6% 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 95.2%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 24.7

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

5. Applied rewrites 24.7%

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

Alternative 17: 13.3% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 kx around 0

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

   1. lower-sin.f64 24.7

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

5. Applied rewrites 24.7%

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

6. Taylor expanded in th around 0

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

8. Applied rewrites 15.5%

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

10. Applied rewrites 15.5%

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

11. Final simplification 15.5%

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

Reproduce

herbie shell --seed 2024272 
(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)))