Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 17.5s

Alternatives: 20

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. +-commutative 95.1%

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

   2. unpow2 95.1%

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

   3. unpow2 95.1%

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

   4. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 41.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-133}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-69}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (fabs (/ (* (sin ky) th) (sin kx)))
   (if (<= (sin kx) 1e-133)
     (sin th)
     (if (<= (sin kx) 1e-69)
       (/ (* (sin ky) (sin th)) (sin ky))
       (/ (sin ky) (/ (sin kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = fabs(((sin(ky) * th) / sin(kx)));
	} else if (sin(kx) <= 1e-133) {
		tmp = sin(th);
	} else if (sin(kx) <= 1e-69) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(ky) / (sin(kx) / 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(kx) <= (-0.4d0)) then
        tmp = abs(((sin(ky) * th) / sin(kx)))
    else if (sin(kx) <= 1d-133) then
        tmp = sin(th)
    else if (sin(kx) <= 1d-69) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = Math.abs(((Math.sin(ky) * th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-133) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 1e-69) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = math.fabs(((math.sin(ky) * th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-133:
		tmp = math.sin(th)
	elif math.sin(kx) <= 1e-69:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = abs(Float64(Float64(sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-133)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-69)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = abs(((sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-133)
		tmp = sin(th);
	elseif (sin(kx) <= 1e-69)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[Abs[N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-133], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-69], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.0%

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

   6. Taylor expanded in th around 0 16.2%

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

      1. add-sqr-sqrt 14.9%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

      4. associate-/l* 22.3%

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

   8. Applied egg-rr 22.3%

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

      1. unpow2 22.3%

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

      2. rem-sqrt-square 28.7%

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

      3. *-commutative 28.7%

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

      4. associate-*l/ 28.7%

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

   10. Simplified 28.7%

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

   if -0.40000000000000002 < (sin.f64 kx) < 1.0000000000000001e-133

   1. Initial program 89.7%

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

      1. unpow2 89.7%

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

      2. sqr-neg 89.7%

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

      3. sin-neg 89.7%

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

      4. sin-neg 89.7%

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

      5. unpow2 89.7%

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

      6. associate-*l/ 87.2%

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

      7. associate-/l* 89.7%

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

      8. unpow2 89.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.5%

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

   if 1.0000000000000001e-133 < (sin.f64 kx) < 9.9999999999999996e-70

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 99.9%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in kx around 0 39.9%

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

   if 9.9999999999999996e-70 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 54.2%

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

      1. clear-num 54.2%

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

      2. un-div-inv 54.2%

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

   7. Applied egg-rr 54.2%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-133}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-69}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{elif}\;\sin ky \leq 10^{-15}:\\ \;\;\;\;\frac{ky \cdot \sin th}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.05)
     (* (/ (sin ky) t_1) th)
     (if (<= (sin ky) 1e-15) (/ (* ky (sin th)) t_1) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.05) {
		tmp = (sin(ky) / t_1) * th;
	} else if (sin(ky) <= 1e-15) {
		tmp = (ky * sin(th)) / t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -0.05) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (Math.sin(ky) <= 1e-15) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.05:
		tmp = (math.sin(ky) / t_1) * th
	elif math.sin(ky) <= 1e-15:
		tmp = (ky * math.sin(th)) / t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.05)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (sin(ky) <= 1e-15)
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.05)
		tmp = (sin(ky) / t_1) * th;
	elseif (sin(ky) <= 1e-15)
		tmp = (ky * sin(th)) / t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-15], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 44.0%

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

      1. *-un-lft-identity 44.0%

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

      2. sqrt-div 44.0%

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

      3. metadata-eval 44.0%

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

      4. +-commutative 44.0%

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

      5. unpow2 44.0%

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

      6. unpow2 44.0%

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

      7. hypot-undefine 44.0%

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

   7. Applied egg-rr 44.0%

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

      1. *-lft-identity 44.0%

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

   9. Simplified 44.0%

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

       1. *-commutative 44.0%

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

       2. un-div-inv 44.0%

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

       3. associate-*l/ 44.0%

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

   11. Applied egg-rr 44.0%

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

       1. associate-*r/ 44.1%

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

   13. Simplified 44.1%

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

   if -0.050000000000000003 < (sin.f64 ky) < 1.0000000000000001e-15

   1. Initial program 90.1%

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

      1. unpow2 90.1%

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

      2. sqr-neg 90.1%

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

      3. sin-neg 90.1%

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

      4. sin-neg 90.1%

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

      5. unpow2 90.1%

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

      6. associate-*l/ 87.7%

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

      7. associate-/l* 90.0%

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

      8. unpow2 90.0%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in ky around 0 95.2%

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

   if 1.0000000000000001e-15 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 57.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin ky \leq 10^{-15}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-91}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (fabs (/ (* (sin ky) th) (sin kx)))
   (if (<= (sin kx) 1e-91) (sin th) (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = fabs(((sin(ky) * th) / sin(kx)));
	} else if (sin(kx) <= 1e-91) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	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(kx) <= (-0.4d0)) then
        tmp = abs(((sin(ky) * th) / sin(kx)))
    else if (sin(kx) <= 1d-91) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = Math.abs(((Math.sin(ky) * th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-91) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = math.fabs(((math.sin(ky) * th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-91:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = abs(Float64(Float64(sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-91)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = abs(((sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-91)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[Abs[N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-91], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.0%

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

   6. Taylor expanded in th around 0 16.2%

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

      1. add-sqr-sqrt 14.9%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

      4. associate-/l* 22.3%

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

   8. Applied egg-rr 22.3%

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

      1. unpow2 22.3%

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

      2. rem-sqrt-square 28.7%

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

      3. *-commutative 28.7%

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

      4. associate-*l/ 28.7%

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

   10. Simplified 28.7%

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

   if -0.40000000000000002 < (sin.f64 kx) < 1.00000000000000002e-91

   1. Initial program 90.3%

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

      1. unpow2 90.3%

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

      2. sqr-neg 90.3%

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

      3. sin-neg 90.3%

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

      4. sin-neg 90.3%

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

      5. unpow2 90.3%

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

      6. associate-*l/ 87.8%

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

      7. associate-/l* 90.2%

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

      8. unpow2 90.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.5%

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

   if 1.00000000000000002e-91 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 53.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-91}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-91}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (fabs (/ (* (sin ky) th) (sin kx)))
   (if (<= (sin kx) 1e-91) (sin th) (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = fabs(((sin(ky) * th) / sin(kx)));
	} else if (sin(kx) <= 1e-91) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	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(kx) <= (-0.4d0)) then
        tmp = abs(((sin(ky) * th) / sin(kx)))
    else if (sin(kx) <= 1d-91) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = Math.abs(((Math.sin(ky) * th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-91) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = math.fabs(((math.sin(ky) * th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-91:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = abs(Float64(Float64(sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-91)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = abs(((sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-91)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[Abs[N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-91], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.0%

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

   6. Taylor expanded in th around 0 16.2%

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

      1. add-sqr-sqrt 14.9%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

      4. associate-/l* 22.3%

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

   8. Applied egg-rr 22.3%

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

      1. unpow2 22.3%

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

      2. rem-sqrt-square 28.7%

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

      3. *-commutative 28.7%

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

      4. associate-*l/ 28.7%

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

   10. Simplified 28.7%

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

   if -0.40000000000000002 < (sin.f64 kx) < 1.00000000000000002e-91

   1. Initial program 90.3%

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

      1. unpow2 90.3%

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

      2. sqr-neg 90.3%

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

      3. sin-neg 90.3%

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

      4. sin-neg 90.3%

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

      5. unpow2 90.3%

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

      6. associate-*l/ 87.8%

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

      7. associate-/l* 90.2%

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

      8. unpow2 90.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.5%

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

   if 1.00000000000000002e-91 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-undefine 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0 53.6%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-91}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-91}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (fabs (/ (* (sin ky) th) (sin kx)))
   (if (<= (sin kx) 1e-91) (sin th) (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = fabs(((sin(ky) * th) / sin(kx)));
	} else if (sin(kx) <= 1e-91) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) / (sin(kx) / 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(kx) <= (-0.4d0)) then
        tmp = abs(((sin(ky) * th) / sin(kx)))
    else if (sin(kx) <= 1d-91) then
        tmp = sin(th)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = Math.abs(((Math.sin(ky) * th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-91) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = math.fabs(((math.sin(ky) * th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-91:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = abs(Float64(Float64(sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-91)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = abs(((sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 1e-91)
		tmp = sin(th);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[Abs[N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-91], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.0%

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

   6. Taylor expanded in th around 0 16.2%

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

      1. add-sqr-sqrt 14.9%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

      4. associate-/l* 22.3%

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

   8. Applied egg-rr 22.3%

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

      1. unpow2 22.3%

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

      2. rem-sqrt-square 28.7%

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

      3. *-commutative 28.7%

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

      4. associate-*l/ 28.7%

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

   10. Simplified 28.7%

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

   if -0.40000000000000002 < (sin.f64 kx) < 1.00000000000000002e-91

   1. Initial program 90.3%

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

      1. unpow2 90.3%

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

      2. sqr-neg 90.3%

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

      3. sin-neg 90.3%

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

      4. sin-neg 90.3%

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

      5. unpow2 90.3%

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

      6. associate-*l/ 87.8%

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

      7. associate-/l* 90.2%

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

      8. unpow2 90.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.5%

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

   if 1.00000000000000002e-91 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 53.7%

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

      1. clear-num 53.6%

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

      2. un-div-inv 53.7%

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

   7. Applied egg-rr 53.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-91}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.282:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-69}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.282)
   (fabs (* (sin ky) (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-69)
     (/ (* (sin ky) (sin th)) (sin ky))
     (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.282) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 1e-69) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(ky) / (sin(kx) / 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(kx) <= (-0.282d0)) then
        tmp = abs((sin(ky) * (sin(th) / sin(kx))))
    else if (sin(kx) <= 1d-69) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.282) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 1e-69) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.282:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 1e-69:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.282)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-69)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.282)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-69)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.282], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-69], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.281999999999999973

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.9%

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

      1. associate-*r/ 16.9%

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

      2. *-commutative 16.9%

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

      3. sin-mult 18.5%

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

      4. associate-/l/ 18.5%

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

      5. +-commutative 18.5%

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

   7. Applied egg-rr 18.5%

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

      1. *-commutative 18.5%

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

      2. associate-/r* 18.5%

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

      3. +-commutative 18.5%

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

      4. sin-mult 16.9%

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

      5. associate-*r/ 16.9%

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

      6. add-sqr-sqrt 15.6%

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

      7. sqrt-unprod 25.8%

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

      8. pow2 25.8%

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

   9. Applied egg-rr 25.8%

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

       1. unpow2 25.8%

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

       2. rem-sqrt-square 37.2%

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

       3. *-commutative 37.2%

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

       4. associate-*l/ 37.2%

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

       5. associate-/l* 37.2%

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

   11. Simplified 37.2%

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

   if -0.281999999999999973 < (sin.f64 kx) < 9.9999999999999996e-70

   1. Initial program 90.5%

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

      1. unpow2 90.5%

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

      2. sqr-neg 90.5%

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

      3. sin-neg 90.5%

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

      4. sin-neg 90.5%

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

      5. unpow2 90.5%

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

      6. associate-*l/ 88.1%

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

      7. associate-/l* 90.4%

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

      8. unpow2 90.4%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in kx around 0 39.6%

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

   if 9.9999999999999996e-70 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 54.2%

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

      1. clear-num 54.2%

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

      2. un-div-inv 54.2%

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

   7. Applied egg-rr 54.2%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.282:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-69}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.28:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-69}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.28)
   (* (sin ky) (fabs (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-69)
     (/ (* (sin ky) (sin th)) (sin ky))
     (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.28) {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	} else if (sin(kx) <= 1e-69) {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	} else {
		tmp = sin(ky) / (sin(kx) / 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(kx) <= (-0.28d0)) then
        tmp = sin(ky) * abs((sin(th) / sin(kx)))
    else if (sin(kx) <= 1d-69) then
        tmp = (sin(ky) * sin(th)) / sin(ky)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.28) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-69) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.28:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	elif math.sin(kx) <= 1e-69:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.28)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-69)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.28)
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	elseif (sin(kx) <= 1e-69)
		tmp = (sin(ky) * sin(th)) / sin(ky);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.28], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-69], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.28000000000000003

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.6%

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

      1. add-sqr-sqrt 5.1%

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

      2. sqrt-unprod 38.6%

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

      3. pow2 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. unpow2 38.6%

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

      2. rem-sqrt-square 42.1%

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

   9. Simplified 42.1%

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

   if -0.28000000000000003 < (sin.f64 kx) < 9.9999999999999996e-70

   1. Initial program 90.4%

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

      1. unpow2 90.4%

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

      2. sqr-neg 90.4%

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

      3. sin-neg 90.4%

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

      4. sin-neg 90.4%

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

      5. unpow2 90.4%

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

      6. associate-*l/ 88.0%

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

      7. associate-/l* 90.3%

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

      8. unpow2 90.3%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in kx around 0 39.9%

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

   if 9.9999999999999996e-70 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 54.2%

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

      1. clear-num 54.2%

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

      2. un-div-inv 54.2%

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

   7. Applied egg-rr 54.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.28:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-69}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. unpow2 95.1%

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

   2. sqr-neg 95.1%

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

   3. sin-neg 95.1%

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

   4. sin-neg 95.1%

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

   5. unpow2 95.1%

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

   6. associate-*l/ 94.0%

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

   7. associate-/l* 95.1%

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

   8. unpow2 95.1%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 10: 38.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (* ky (fabs (/ th (sin kx))))
   (if (<= (sin kx) 5e-69) (sin th) (* ky (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = ky * fabs((th / sin(kx)));
	} else if (sin(kx) <= 5e-69) {
		tmp = sin(th);
	} else {
		tmp = ky * (sin(th) / sin(kx));
	}
	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(kx) <= (-0.4d0)) then
        tmp = ky * abs((th / sin(kx)))
    else if (sin(kx) <= 5d-69) then
        tmp = sin(th)
    else
        tmp = ky * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = ky * Math.abs((th / Math.sin(kx)));
	} else if (Math.sin(kx) <= 5e-69) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = ky * math.fabs((th / math.sin(kx)))
	elif math.sin(kx) <= 5e-69:
		tmp = math.sin(th)
	else:
		tmp = ky * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = Float64(ky * abs(Float64(th / sin(kx))));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = ky * abs((th / sin(kx)));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = ky * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[(ky * N[Abs[N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-69], N[Sin[th], $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 62.5%

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

   6. Taylor expanded in ky around 0 16.4%

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

      1. associate-/l* 16.4%

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

   8. Simplified 16.4%

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

      1. add-sqr-sqrt 3.1%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

   10. Applied egg-rr 26.6%

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

       1. unpow2 26.6%

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

       2. rem-sqrt-square 28.3%

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

   12. Simplified 28.3%

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

   if -0.40000000000000002 < (sin.f64 kx) < 5.00000000000000033e-69

   1. Initial program 90.9%

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

      1. unpow2 90.9%

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

      2. sqr-neg 90.9%

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

      3. sin-neg 90.9%

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

      4. sin-neg 90.9%

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

      5. unpow2 90.9%

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

      6. associate-*l/ 88.5%

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

      7. associate-/l* 90.8%

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

      8. unpow2 90.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.1%

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

   if 5.00000000000000033e-69 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 46.2%

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

      1. associate-/l* 46.2%

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

   7. Simplified 46.2%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (* ky (fabs (/ th (sin kx))))
   (if (<= (sin kx) 5e-69) (sin th) (* (sin th) (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = ky * fabs((th / sin(kx)));
	} else if (sin(kx) <= 5e-69) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	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(kx) <= (-0.4d0)) then
        tmp = ky * abs((th / sin(kx)))
    else if (sin(kx) <= 5d-69) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = ky * Math.abs((th / Math.sin(kx)));
	} else if (Math.sin(kx) <= 5e-69) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = ky * math.fabs((th / math.sin(kx)))
	elif math.sin(kx) <= 5e-69:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = Float64(ky * abs(Float64(th / sin(kx))));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = ky * abs((th / sin(kx)));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[(ky * N[Abs[N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-69], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 62.5%

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

   6. Taylor expanded in ky around 0 16.4%

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

      1. associate-/l* 16.4%

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

   8. Simplified 16.4%

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

      1. add-sqr-sqrt 3.1%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

   10. Applied egg-rr 26.6%

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

       1. unpow2 26.6%

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

       2. rem-sqrt-square 28.3%

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

   12. Simplified 28.3%

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

   if -0.40000000000000002 < (sin.f64 kx) < 5.00000000000000033e-69

   1. Initial program 90.9%

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

      1. unpow2 90.9%

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

      2. sqr-neg 90.9%

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

      3. sin-neg 90.9%

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

      4. sin-neg 90.9%

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

      5. unpow2 90.9%

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

      6. associate-*l/ 88.5%

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

      7. associate-/l* 90.8%

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

      8. unpow2 90.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.1%

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

   if 5.00000000000000033e-69 < (sin.f64 kx)

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0 46.1%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (* ky (fabs (/ th (sin kx))))
   (if (<= (sin kx) 5e-69) (sin th) (/ (* ky (sin th)) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = ky * fabs((th / sin(kx)));
	} else if (sin(kx) <= 5e-69) {
		tmp = sin(th);
	} else {
		tmp = (ky * sin(th)) / sin(kx);
	}
	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(kx) <= (-0.4d0)) then
        tmp = ky * abs((th / sin(kx)))
    else if (sin(kx) <= 5d-69) then
        tmp = sin(th)
    else
        tmp = (ky * sin(th)) / sin(kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = ky * Math.abs((th / Math.sin(kx)));
	} else if (Math.sin(kx) <= 5e-69) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = ky * math.fabs((th / math.sin(kx)))
	elif math.sin(kx) <= 5e-69:
		tmp = math.sin(th)
	else:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = Float64(ky * abs(Float64(th / sin(kx))));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = ky * abs((th / sin(kx)));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = (ky * sin(th)) / sin(kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[(ky * N[Abs[N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-69], N[Sin[th], $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 62.5%

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

   6. Taylor expanded in ky around 0 16.4%

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

      1. associate-/l* 16.4%

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

   8. Simplified 16.4%

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

      1. add-sqr-sqrt 3.1%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

   10. Applied egg-rr 26.6%

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

       1. unpow2 26.6%

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

       2. rem-sqrt-square 28.3%

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

   12. Simplified 28.3%

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

   if -0.40000000000000002 < (sin.f64 kx) < 5.00000000000000033e-69

   1. Initial program 90.9%

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

      1. unpow2 90.9%

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

      2. sqr-neg 90.9%

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

      3. sin-neg 90.9%

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

      4. sin-neg 90.9%

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

      5. unpow2 90.9%

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

      6. associate-*l/ 88.5%

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

      7. associate-/l* 90.8%

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

      8. unpow2 90.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.1%

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

   if 5.00000000000000033e-69 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 46.2%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.4)
   (fabs (/ (* (sin ky) th) (sin kx)))
   (if (<= (sin kx) 5e-69) (sin th) (/ (* ky (sin th)) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.4) {
		tmp = fabs(((sin(ky) * th) / sin(kx)));
	} else if (sin(kx) <= 5e-69) {
		tmp = sin(th);
	} else {
		tmp = (ky * sin(th)) / sin(kx);
	}
	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(kx) <= (-0.4d0)) then
        tmp = abs(((sin(ky) * th) / sin(kx)))
    else if (sin(kx) <= 5d-69) then
        tmp = sin(th)
    else
        tmp = (ky * sin(th)) / sin(kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.4) {
		tmp = Math.abs(((Math.sin(ky) * th) / Math.sin(kx)));
	} else if (Math.sin(kx) <= 5e-69) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.4:
		tmp = math.fabs(((math.sin(ky) * th) / math.sin(kx)))
	elif math.sin(kx) <= 5e-69:
		tmp = math.sin(th)
	else:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.4)
		tmp = abs(Float64(Float64(sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.4)
		tmp = abs(((sin(ky) * th) / sin(kx)));
	elseif (sin(kx) <= 5e-69)
		tmp = sin(th);
	else
		tmp = (ky * sin(th)) / sin(kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.4], N[Abs[N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-69], N[Sin[th], $MachinePrecision], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 16.0%

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

   6. Taylor expanded in th around 0 16.2%

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

      1. add-sqr-sqrt 14.9%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

      4. associate-/l* 22.3%

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

   8. Applied egg-rr 22.3%

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

      1. unpow2 22.3%

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

      2. rem-sqrt-square 28.7%

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

      3. *-commutative 28.7%

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

      4. associate-*l/ 28.7%

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

   10. Simplified 28.7%

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

   if -0.40000000000000002 < (sin.f64 kx) < 5.00000000000000033e-69

   1. Initial program 90.9%

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

      1. unpow2 90.9%

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

      2. sqr-neg 90.9%

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

      3. sin-neg 90.9%

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

      4. sin-neg 90.9%

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

      5. unpow2 90.9%

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

      6. associate-*l/ 88.5%

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

      7. associate-/l* 90.8%

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

      8. unpow2 90.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 34.1%

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

   if 5.00000000000000033e-69 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 46.2%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.4:\\ \;\;\;\;\left|\frac{\sin ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.8:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;th \leq 5 \cdot 10^{+76} \lor \neg \left(th \leq 3 \cdot 10^{+121}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.8)
   (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
   (if (or (<= th 5e+76) (not (<= th 3e+121)))
     (sin th)
     (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.8) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if ((th <= 5e+76) || !(th <= 3e+121)) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.8) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else if ((th <= 5e+76) || !(th <= 3e+121)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.8:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	elif (th <= 5e+76) or not (th <= 3e+121):
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.8)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif ((th <= 5e+76) || !(th <= 3e+121))
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.8)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	elseif ((th <= 5e+76) || ~((th <= 3e+121)))
		tmp = sin(th);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[Or[LessEqual[th, 5e+76], N[Not[LessEqual[th, 3e+121]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 0.80000000000000004

   1. Initial program 95.6%

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

      1. unpow2 95.6%

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

      2. sqr-neg 95.6%

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

      3. sin-neg 95.6%

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

      4. sin-neg 95.6%

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

      5. unpow2 95.6%

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

      6. associate-*l/ 94.0%

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

      7. associate-/l* 95.5%

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

      8. unpow2 95.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 65.9%

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

      1. *-un-lft-identity 65.9%

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

      2. sqrt-div 66.4%

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

      3. metadata-eval 66.4%

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

      4. +-commutative 66.4%

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

      5. unpow2 66.4%

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

      6. unpow2 66.4%

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

      7. hypot-undefine 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. *-lft-identity 68.2%

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

   9. Simplified 68.2%

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

       1. *-commutative 68.2%

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

       2. un-div-inv 68.3%

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

       3. associate-*l/ 66.4%

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

   11. Applied egg-rr 66.4%

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

       1. associate-*r/ 68.4%

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

   13. Simplified 68.4%

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

   if 0.80000000000000004 < th < 4.99999999999999991e76 or 3.0000000000000002e121 < th

   1. Initial program 94.5%

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

      1. unpow2 94.5%

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

      2. sqr-neg 94.5%

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

      3. sin-neg 94.5%

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

      4. sin-neg 94.5%

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

      5. unpow2 94.5%

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

      6. associate-*l/ 94.6%

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

      7. associate-/l* 94.5%

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

      8. unpow2 94.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 21.6%

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

   if 4.99999999999999991e76 < th < 3.0000000000000002e121

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. sqr-neg 91.4%

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

      3. sin-neg 91.4%

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

      4. sin-neg 91.4%

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

      5. unpow2 91.4%

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

      6. associate-*l/ 91.4%

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

      7. associate-/l* 91.7%

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

      8. unpow2 91.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 26.8%

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

      1. clear-num 26.8%

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

      2. un-div-inv 26.8%

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

   7. Applied egg-rr 26.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.8:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;th \leq 5 \cdot 10^{+76} \lor \neg \left(th \leq 3 \cdot 10^{+121}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.2 \cdot 10^{-20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.2e-20) (sin th) (* ky (fabs (/ th (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.2e-20) {
		tmp = sin(th);
	} else {
		tmp = ky * fabs((th / sin(kx)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 6.2d-20) then
        tmp = sin(th)
    else
        tmp = ky * abs((th / sin(kx)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.2e-20) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * Math.abs((th / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.2e-20:
		tmp = math.sin(th)
	else:
		tmp = ky * math.fabs((th / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.2e-20)
		tmp = sin(th);
	else
		tmp = Float64(ky * abs(Float64(th / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.2e-20)
		tmp = sin(th);
	else
		tmp = ky * abs((th / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.2e-20], N[Sin[th], $MachinePrecision], N[(ky * N[Abs[N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.19999999999999999e-20

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 91.9%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 27.5%

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

   if 6.19999999999999999e-20 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 54.7%

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

   6. Taylor expanded in ky around 0 23.6%

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

      1. associate-/l* 23.5%

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

   8. Simplified 23.5%

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

      1. add-sqr-sqrt 8.1%

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

      2. sqrt-unprod 21.2%

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

      3. pow2 21.2%

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

   10. Applied egg-rr 21.2%

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

       1. unpow2 21.2%

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

       2. rem-sqrt-square 23.7%

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

   12. Simplified 23.7%

       \[\leadsto ky \cdot \color{blue}{\left|\frac{th}{\sin kx}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 6.2 \cdot 10^{-20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \left|\frac{th}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6 \cdot 10^{-20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6e-20) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6e-20) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 6d-20) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6e-20) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6e-20:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6e-20)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6e-20)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6e-20], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.00000000000000057e-20

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 91.9%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 27.5%

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

   if 6.00000000000000057e-20 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 54.7%

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

   6. Taylor expanded in ky around 0 23.6%

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

      1. associate-/l* 23.5%

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

   8. Simplified 23.5%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 6 \cdot 10^{-20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.3 \cdot 10^{-20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.3e-20) (sin th) (/ ky (/ (sin kx) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.3e-20) {
		tmp = sin(th);
	} else {
		tmp = ky / (sin(kx) / th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 4.3d-20) then
        tmp = sin(th)
    else
        tmp = ky / (sin(kx) / th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.3e-20) {
		tmp = Math.sin(th);
	} else {
		tmp = ky / (Math.sin(kx) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.3e-20:
		tmp = math.sin(th)
	else:
		tmp = ky / (math.sin(kx) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.3e-20)
		tmp = sin(th);
	else
		tmp = Float64(ky / Float64(sin(kx) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.3e-20)
		tmp = sin(th);
	else
		tmp = ky / (sin(kx) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.3e-20], N[Sin[th], $MachinePrecision], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 4.30000000000000011e-20

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 91.9%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 27.5%

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

   if 4.30000000000000011e-20 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 54.7%

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

   6. Taylor expanded in ky around 0 23.6%

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

      1. associate-/l* 23.5%

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

   8. Simplified 23.5%

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

      1. clear-num 23.5%

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

      2. un-div-inv 23.6%

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

   10. Applied egg-rr 23.6%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.3 \cdot 10^{-20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 24.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 10^{-152}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1e-152) (* ky (/ th kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1e-152) {
		tmp = ky * (th / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1d-152) then
        tmp = ky * (th / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1e-152) {
		tmp = ky * (th / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1e-152:
		tmp = ky * (th / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1e-152)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1e-152)
		tmp = ky * (th / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1e-152], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.00000000000000007e-152

   1. Initial program 92.7%

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

      1. unpow2 92.7%

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

      2. sqr-neg 92.7%

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

      3. sin-neg 92.7%

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

      4. sin-neg 92.7%

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

      5. unpow2 92.7%

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

      6. associate-*l/ 91.6%

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

      7. associate-/l* 92.7%

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

      8. unpow2 92.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 49.6%

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

   6. Taylor expanded in ky around 0 22.7%

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

      1. associate-/l* 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in kx around 0 22.9%

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

   if 1.00000000000000007e-152 < ky

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 98.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 30.7%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 10^{-152}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 18.4% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.55 \cdot 10^{-42}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.55e-42) (* ky (/ th kx)) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.55e-42) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.55d-42) then
        tmp = ky * (th / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.55e-42) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.55e-42:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.55e-42)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.55e-42)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.55e-42], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 1.5500000000000001e-42

   1. Initial program 93.5%

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

      1. unpow2 93.5%

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

      2. sqr-neg 93.5%

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

      3. sin-neg 93.5%

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

      4. sin-neg 93.5%

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

      5. unpow2 93.5%

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

      6. associate-*l/ 92.0%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 49.2%

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

   6. Taylor expanded in ky around 0 23.2%

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

      1. associate-/l* 24.0%

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

   8. Simplified 24.0%

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

   9. Taylor expanded in kx around 0 22.3%

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

   if 1.5500000000000001e-42 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in th around 0 48.1%

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

   6. Taylor expanded in kx around 0 18.2%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.55 \cdot 10^{-42}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 13.8% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

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

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 95.1%

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

   1. unpow2 95.1%

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

   2. sqr-neg 95.1%

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

   3. sin-neg 95.1%

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

   4. sin-neg 95.1%

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

   5. unpow2 95.1%

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

   6. associate-*l/ 94.0%

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

   7. associate-/l* 95.1%

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

   8. unpow2 95.1%

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

3. Simplified 99.7%

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

5. Taylor expanded in th around 0 48.9%

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

6. Taylor expanded in kx around 0 14.2%

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

7. Final simplification 14.2%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))