Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 34.2s

Alternatives: 19

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: 94.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. +-commutative 93.4%

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

   2. unpow2 93.4%

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

   3. unpow2 93.4%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 94.8%

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

   2. associate-*r/ 99.5%

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

   3. expm1-log1p-u 99.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

   4. expm1-udef 40.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

5. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 99.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

   2. expm1-log1p 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-/r/ 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-49)
   (fabs (sin th))
   (if (<= (sin ky) -1e-304)
     (/ (sin ky) (/ (sin kx) (sin th)))
     (if (<= (sin ky) 2e-57)
       (* (sin th) (fabs (/ (sin ky) (sin kx))))
       (/ (* (sin th) (sin ky)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-49) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -1e-304) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (sin(ky) <= 2e-57) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else {
		tmp = (sin(th) * sin(ky)) / sin(ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-49)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-1d-304)) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else if (sin(ky) <= 2d-57) then
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    else
        tmp = (sin(th) * sin(ky)) / sin(ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-49) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -1e-304) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (Math.sin(ky) <= 2e-57) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-49:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -1e-304:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif math.sin(ky) <= 2e-57:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	else:
		tmp = (math.sin(th) * math.sin(ky)) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -1e-304)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (sin(ky) <= 2e-57)
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	else
		tmp = Float64(Float64(sin(th) * sin(ky)) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -1e-304)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (sin(ky) <= 2e-57)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	else
		tmp = (sin(th) * sin(ky)) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-49], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-304], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-57], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -9.99999999999999936e-50

   1. Initial program 99.5%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 28.2%

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

      3. pow2 28.2%

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

      4. clear-num 28.1%

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

      5. un-div-inv 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. rem-sqrt-square 32.6%

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

      3. associate-/r/ 32.7%

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

      4. *-inverses 32.7%

         \[\leadsto \left|\color{blue}{1} \cdot \sin th\right| \]

      5. *-lft-identity 32.7%

         \[\leadsto \left|\color{blue}{\sin th}\right| \]

   8. Simplified 32.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -9.99999999999999936e-50 < (sin.f64 ky) < -9.99999999999999971e-305

   1. Initial program 87.0%

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

      1. associate-/r/ 87.1%

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

      2. +-commutative 87.1%

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

      3. unpow2 87.1%

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

      4. unpow2 87.1%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 44.3%

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

   if -9.99999999999999971e-305 < (sin.f64 ky) < 1.99999999999999991e-57

   1. Initial program 87.3%

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

      1. +-commutative 87.3%

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

      2. unpow2 87.3%

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

      3. unpow2 87.3%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 44.6%

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

      1. add-sqr-sqrt 37.0%

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

      2. sqrt-unprod 51.7%

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

      3. pow2 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unpow2 51.7%

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

      2. rem-sqrt-square 78.9%

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

   8. Simplified 78.9%

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

   if 1.99999999999999991e-57 < (sin.f64 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. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in kx around 0 56.7%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \]

Alternative 3: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.004:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.004)
   (/ (sin ky) (/ (sin kx) (sin th)))
   (if (<= (sin th) 1e-7)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (* (sin ky) (fabs (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.004) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (sin(th) <= 1e-7) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.004) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (Math.sin(th) <= 1e-7) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.004:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif math.sin(th) <= 1e-7:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.004)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (sin(th) <= 1e-7)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.004)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (sin(th) <= 1e-7)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.004], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-7], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -0.0040000000000000001

   1. Initial program 92.0%

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

      1. associate-/r/ 92.0%

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

      2. +-commutative 92.0%

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

      3. unpow2 92.0%

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

      4. unpow2 92.0%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 23.3%

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

   if -0.0040000000000000001 < (sin.f64 th) < 9.9999999999999995e-8

   1. Initial program 94.0%

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

      1. associate-*l/ 87.5%

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

      2. +-commutative 87.5%

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

      3. unpow2 87.5%

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

      4. unpow2 87.5%

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

      5. hypot-def 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in th around 0 89.3%

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

      1. expm1-log1p-u 89.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 19.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 19.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 19.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 19.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 19.5%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 99.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l/ 89.3%

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

      5. associate-*r/ 99.6%

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

   8. Simplified 99.6%

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

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

   1. Initial program 94.1%

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

      1. associate-*l/ 94.2%

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

      2. associate-*r/ 94.1%

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

      3. +-commutative 94.1%

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

      4. unpow2 94.1%

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

      5. unpow2 94.1%

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

      6. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 27.2%

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

      1. add-sqr-sqrt 25.5%

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

      2. sqrt-unprod 54.4%

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

      3. pow2 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. unpow2 54.4%

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

      2. rem-sqrt-square 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.004:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]

Alternative 4: 48.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-49)
   (fabs (sin th))
   (if (<= (sin ky) 2e-154)
     (/ (sin th) (/ (sin kx) ky))
     (/ (* (sin th) (sin ky)) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-49) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-154) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = (sin(th) * sin(ky)) / sin(ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-49)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-154) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = (sin(th) * sin(ky)) / sin(ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-49) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-154) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-49:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-154:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = (math.sin(th) * math.sin(ky)) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-154)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = Float64(Float64(sin(th) * sin(ky)) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-154)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = (sin(th) * sin(ky)) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-49], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-154], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.99999999999999936e-50

   1. Initial program 99.5%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 28.2%

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

      3. pow2 28.2%

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

      4. clear-num 28.1%

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

      5. un-div-inv 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. rem-sqrt-square 32.6%

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

      3. associate-/r/ 32.7%

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

      4. *-inverses 32.7%

         \[\leadsto \left|\color{blue}{1} \cdot \sin th\right| \]

      5. *-lft-identity 32.7%

         \[\leadsto \left|\color{blue}{\sin th}\right| \]

   8. Simplified 32.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -9.99999999999999936e-50 < (sin.f64 ky) < 1.9999999999999999e-154

   1. Initial program 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. unpow2 85.2%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 44.1%

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

      1. associate-/l* 45.6%

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

   6. Simplified 45.6%

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

   if 1.9999999999999999e-154 < (sin.f64 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. associate-*l/ 98.3%

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

      2. +-commutative 98.3%

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

      3. unpow2 98.3%

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

      4. unpow2 98.3%

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

      5. hypot-def 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in kx around 0 54.1%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \]

Alternative 5: 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 93.4%

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

   1. associate-*l/ 90.4%

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

   2. associate-*r/ 93.4%

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

   3. +-commutative 93.4%

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

   4. unpow2 93.4%

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

   5. unpow2 93.4%

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

   6. hypot-def 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. +-commutative 93.4%

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

   2. unpow2 93.4%

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

   3. unpow2 93.4%

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

   4. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 7: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -3000000000000 \lor \neg \left(ky \leq 0.025\right):\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (or (<= ky -3000000000000.0) (not (<= ky 0.025)))
     (* (sin ky) (/ th t_1))
     (* (sin th) (/ ky t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if ((ky <= -3000000000000.0) || !(ky <= 0.025)) {
		tmp = sin(ky) * (th / t_1);
	} else {
		tmp = sin(th) * (ky / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if ((ky <= -3000000000000.0) || !(ky <= 0.025)) {
		tmp = Math.sin(ky) * (th / t_1);
	} else {
		tmp = Math.sin(th) * (ky / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if (ky <= -3000000000000.0) or not (ky <= 0.025):
		tmp = math.sin(ky) * (th / t_1)
	else:
		tmp = math.sin(th) * (ky / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if ((ky <= -3000000000000.0) || !(ky <= 0.025))
		tmp = Float64(sin(ky) * Float64(th / t_1));
	else
		tmp = Float64(sin(th) * Float64(ky / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if ((ky <= -3000000000000.0) || ~((ky <= 0.025)))
		tmp = sin(ky) * (th / t_1);
	else
		tmp = sin(th) * (ky / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[ky, -3000000000000.0], N[Not[LessEqual[ky, 0.025]], $MachinePrecision]], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -3e12 or 0.025000000000000001 < 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. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 55.9%

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

      1. expm1-log1p-u 55.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-udef 5.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

      3. div-inv 5.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-commutative 5.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(th \cdot \sin ky\right)} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. associate-*l* 5.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{th \cdot \left(\sin ky \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}\right)} - 1 \]

      6. div-inv 5.7%

         \[\leadsto e^{\mathsf{log1p}\left(th \cdot \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

   6. Applied egg-rr 5.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-*l/ 55.9%

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

      5. associate-*r/ 56.0%

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

   8. Simplified 56.0%

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

   if -3e12 < ky < 0.025000000000000001

   1. Initial program 88.8%

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

      1. associate-/r/ 88.8%

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

      2. +-commutative 88.8%

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

      3. unpow2 88.8%

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

      4. unpow2 88.8%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 91.3%

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

      2. clear-num 90.8%

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

      3. associate-/r/ 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in ky around 0 89.1%

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

      1. expm1-log1p-u 89.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\sin th \cdot ky\right)\right)\right)} \]

      2. expm1-udef 32.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\sin th \cdot ky\right)\right)} - 1} \]

      3. associate-*l/ 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(\sin th \cdot ky\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. *-un-lft-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      6. times-frac 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin th}{1} \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      7. /-rgt-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th} \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 97.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3000000000000 \lor \neg \left(ky \leq 0.025\right):\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 8: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -3000000000000 \lor \neg \left(ky \leq 0.0148\right):\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (or (<= ky -3000000000000.0) (not (<= ky 0.0148)))
     (* th (/ (sin ky) t_1))
     (* (sin th) (/ ky t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if ((ky <= -3000000000000.0) || !(ky <= 0.0148)) {
		tmp = th * (sin(ky) / t_1);
	} else {
		tmp = sin(th) * (ky / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if ((ky <= -3000000000000.0) || !(ky <= 0.0148)) {
		tmp = th * (Math.sin(ky) / t_1);
	} else {
		tmp = Math.sin(th) * (ky / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if (ky <= -3000000000000.0) or not (ky <= 0.0148):
		tmp = th * (math.sin(ky) / t_1)
	else:
		tmp = math.sin(th) * (ky / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if ((ky <= -3000000000000.0) || !(ky <= 0.0148))
		tmp = Float64(th * Float64(sin(ky) / t_1));
	else
		tmp = Float64(sin(th) * Float64(ky / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if ((ky <= -3000000000000.0) || ~((ky <= 0.0148)))
		tmp = th * (sin(ky) / t_1);
	else
		tmp = sin(th) * (ky / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[ky, -3000000000000.0], N[Not[LessEqual[ky, 0.0148]], $MachinePrecision]], N[(th * N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -3e12 or 0.014800000000000001 < 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. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 55.9%

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

      1. associate-/l* 56.0%

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

      2. associate-/r/ 56.0%

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

   6. Applied egg-rr 56.0%

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

   if -3e12 < ky < 0.014800000000000001

   1. Initial program 88.8%

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

      1. associate-/r/ 88.8%

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

      2. +-commutative 88.8%

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

      3. unpow2 88.8%

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

      4. unpow2 88.8%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 91.3%

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

      2. clear-num 90.8%

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

      3. associate-/r/ 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in ky around 0 89.1%

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

      1. expm1-log1p-u 89.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\sin th \cdot ky\right)\right)\right)} \]

      2. expm1-udef 32.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\sin th \cdot ky\right)\right)} - 1} \]

      3. associate-*l/ 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(\sin th \cdot ky\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. *-un-lft-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      6. times-frac 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin th}{1} \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      7. /-rgt-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th} \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 97.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3000000000000 \lor \neg \left(ky \leq 0.0148\right):\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 9: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -3000000000000:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;ky \leq 0.0077:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= ky -3000000000000.0)
     (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
     (if (<= ky 0.0077) (* (sin th) (/ ky t_1)) (* th (/ (sin ky) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (ky <= -3000000000000.0) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (ky <= 0.0077) {
		tmp = sin(th) * (ky / t_1);
	} else {
		tmp = th * (sin(ky) / t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (ky <= -3000000000000.0) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (ky <= 0.0077) {
		tmp = Math.sin(th) * (ky / t_1);
	} else {
		tmp = th * (Math.sin(ky) / t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if ky <= -3000000000000.0:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif ky <= 0.0077:
		tmp = math.sin(th) * (ky / t_1)
	else:
		tmp = th * (math.sin(ky) / t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (ky <= -3000000000000.0)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (ky <= 0.0077)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	else
		tmp = Float64(th * Float64(sin(ky) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (ky <= -3000000000000.0)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (ky <= 0.0077)
		tmp = sin(th) * (ky / t_1);
	else
		tmp = th * (sin(ky) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[ky, -3000000000000.0], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.0077], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -3e12

   1. Initial program 99.5%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 54.7%

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

      1. associate-*r/ 54.9%

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

      2. unpow2 54.9%

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

      3. unpow2 54.9%

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

      4. hypot-def 54.9%

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

      5. *-rgt-identity 54.9%

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

      6. hypot-def 54.9%

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

      7. unpow2 54.9%

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

      8. unpow2 54.9%

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

      9. +-commutative 54.9%

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

      10. unpow2 54.9%

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

      11. unpow2 54.9%

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

      12. hypot-def 54.9%

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

   6. Simplified 54.9%

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

   if -3e12 < ky < 0.0077000000000000002

   1. Initial program 88.8%

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

      1. associate-/r/ 88.8%

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

      2. +-commutative 88.8%

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

      3. unpow2 88.8%

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

      4. unpow2 88.8%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 91.3%

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

      2. clear-num 90.8%

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

      3. associate-/r/ 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in ky around 0 89.1%

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

      1. expm1-log1p-u 89.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\sin th \cdot ky\right)\right)\right)} \]

      2. expm1-udef 32.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\sin th \cdot ky\right)\right)} - 1} \]

      3. associate-*l/ 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(\sin th \cdot ky\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin th \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

      5. *-un-lft-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sin th \cdot ky}{\color{blue}{1 \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      6. times-frac 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin th}{1} \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} - 1 \]

      7. /-rgt-identity 32.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin th} \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 97.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]

      2. expm1-log1p 97.5%

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

   10. Simplified 97.5%

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

   if 0.0077000000000000002 < 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. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 57.0%

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

      1. associate-/l* 57.1%

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

      2. associate-/r/ 57.2%

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

   6. Applied egg-rr 57.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3000000000000:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;ky \leq 0.0077:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]

Alternative 10: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-49)
   (fabs (sin th))
   (if (<= (sin ky) 2e-154) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-49) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-154) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-49)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-154) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-49) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-154) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-49:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-154:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-154)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-154)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-49], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-154], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.99999999999999936e-50

   1. Initial program 99.5%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 28.2%

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

      3. pow2 28.2%

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

      4. clear-num 28.1%

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

      5. un-div-inv 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. rem-sqrt-square 32.6%

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

      3. associate-/r/ 32.7%

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

      4. *-inverses 32.7%

         \[\leadsto \left|\color{blue}{1} \cdot \sin th\right| \]

      5. *-lft-identity 32.7%

         \[\leadsto \left|\color{blue}{\sin th}\right| \]

   8. Simplified 32.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -9.99999999999999936e-50 < (sin.f64 ky) < 1.9999999999999999e-154

   1. Initial program 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. unpow2 85.2%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 45.5%

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

   if 1.9999999999999999e-154 < (sin.f64 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 53.1%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-49)
   (fabs (sin th))
   (if (<= (sin ky) 2e-154) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-49) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-154) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-49)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-154) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-49) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-154) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-49:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-154:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-154)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-49)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-154)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-49], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-154], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.99999999999999936e-50

   1. Initial program 99.5%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 28.2%

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

      3. pow2 28.2%

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

      4. clear-num 28.1%

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

      5. un-div-inv 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. rem-sqrt-square 32.6%

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

      3. associate-/r/ 32.7%

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

      4. *-inverses 32.7%

         \[\leadsto \left|\color{blue}{1} \cdot \sin th\right| \]

      5. *-lft-identity 32.7%

         \[\leadsto \left|\color{blue}{\sin th}\right| \]

   8. Simplified 32.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -9.99999999999999936e-50 < (sin.f64 ky) < 1.9999999999999999e-154

   1. Initial program 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. unpow2 85.2%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 44.1%

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

      1. associate-/l* 45.6%

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

   6. Simplified 45.6%

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

   if 1.9999999999999999e-154 < (sin.f64 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 53.1%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 33.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-152}:\\ \;\;\;\;\left(kx \cdot 0.16666666666666666 + \frac{1}{kx}\right) \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;ky \leq 22000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.35 \cdot 10^{+272}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -2e-158)
     t_1
     (if (<= ky 3.6e-152)
       (* (+ (* kx 0.16666666666666666) (/ 1.0 kx)) (* (sin th) ky))
       (if (<= ky 22000.0) (sin th) (if (<= ky 2.35e+272) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -2e-158) {
		tmp = t_1;
	} else if (ky <= 3.6e-152) {
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (sin(th) * ky);
	} else if (ky <= 22000.0) {
		tmp = sin(th);
	} else if (ky <= 2.35e+272) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (ky <= (-2d-158)) then
        tmp = t_1
    else if (ky <= 3.6d-152) then
        tmp = ((kx * 0.16666666666666666d0) + (1.0d0 / kx)) * (sin(th) * ky)
    else if (ky <= 22000.0d0) then
        tmp = sin(th)
    else if (ky <= 2.35d+272) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -2e-158) {
		tmp = t_1;
	} else if (ky <= 3.6e-152) {
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (Math.sin(th) * ky);
	} else if (ky <= 22000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 2.35e+272) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -2e-158:
		tmp = t_1
	elif ky <= 3.6e-152:
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (math.sin(th) * ky)
	elif ky <= 22000.0:
		tmp = math.sin(th)
	elif ky <= 2.35e+272:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -2e-158)
		tmp = t_1;
	elseif (ky <= 3.6e-152)
		tmp = Float64(Float64(Float64(kx * 0.16666666666666666) + Float64(1.0 / kx)) * Float64(sin(th) * ky));
	elseif (ky <= 22000.0)
		tmp = sin(th);
	elseif (ky <= 2.35e+272)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -2e-158)
		tmp = t_1;
	elseif (ky <= 3.6e-152)
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (sin(th) * ky);
	elseif (ky <= 22000.0)
		tmp = sin(th);
	elseif (ky <= 2.35e+272)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -2e-158], t$95$1, If[LessEqual[ky, 3.6e-152], N[(N[(N[(kx * 0.16666666666666666), $MachinePrecision] + N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 22000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 2.35e+272], t$95$1, N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -2.00000000000000013e-158 or 22000 < ky < 2.35e272

   1. Initial program 99.3%

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

      1. associate-*l/ 97.9%

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

      2. associate-*r/ 99.2%

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

      3. +-commutative 99.2%

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

      4. unpow2 99.2%

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

      5. unpow2 99.2%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 18.3%

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

      1. add-sqr-sqrt 6.8%

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

      2. sqrt-unprod 24.3%

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

      3. pow2 24.3%

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

      4. clear-num 24.3%

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

      5. un-div-inv 24.3%

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

   6. Applied egg-rr 24.3%

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

      1. unpow2 24.3%

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

      2. rem-sqrt-square 28.0%

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

      3. associate-/r/ 28.0%

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

      4. *-inverses 28.0%

         \[\leadsto \left|\color{blue}{1} \cdot \sin th\right| \]

      5. *-lft-identity 28.0%

         \[\leadsto \left|\color{blue}{\sin th}\right| \]

   8. Simplified 28.0%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -2.00000000000000013e-158 < ky < 3.6e-152

   1. Initial program 79.6%

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

      1. +-commutative 79.6%

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

      2. unpow2 79.6%

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

      3. unpow2 79.6%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 50.2%

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

   5. Taylor expanded in kx around 0 22.0%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\sin ky \cdot kx\right) + \frac{\sin ky}{kx}\right)} \cdot \sin th \]

   6. Taylor expanded in ky around 0 27.2%

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

   if 3.6e-152 < ky < 22000 or 2.35e272 < 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 54.9%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-152}:\\ \;\;\;\;\left(kx \cdot 0.16666666666666666 + \frac{1}{kx}\right) \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;ky \leq 22000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.35 \cdot 10^{+272}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 32.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -180:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.7 \cdot 10^{-151}:\\ \;\;\;\;\left(kx \cdot 0.16666666666666666 + \frac{1}{kx}\right) \cdot \left(\sin th \cdot ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -180.0)
   (sin th)
   (if (<= ky 4.7e-151)
     (* (+ (* kx 0.16666666666666666) (/ 1.0 kx)) (* (sin th) ky))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -180.0) {
		tmp = sin(th);
	} else if (ky <= 4.7e-151) {
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (sin(th) * ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-180.0d0)) then
        tmp = sin(th)
    else if (ky <= 4.7d-151) then
        tmp = ((kx * 0.16666666666666666d0) + (1.0d0 / kx)) * (sin(th) * ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -180.0) {
		tmp = Math.sin(th);
	} else if (ky <= 4.7e-151) {
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (Math.sin(th) * ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -180.0:
		tmp = math.sin(th)
	elif ky <= 4.7e-151:
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (math.sin(th) * ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -180.0)
		tmp = sin(th);
	elseif (ky <= 4.7e-151)
		tmp = Float64(Float64(Float64(kx * 0.16666666666666666) + Float64(1.0 / kx)) * Float64(sin(th) * ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -180.0)
		tmp = sin(th);
	elseif (ky <= 4.7e-151)
		tmp = ((kx * 0.16666666666666666) + (1.0 / kx)) * (sin(th) * ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -180.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 4.7e-151], N[(N[(N[(kx * 0.16666666666666666), $MachinePrecision] + N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -180 or 4.70000000000000029e-151 < 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. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 32.7%

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

   if -180 < ky < 4.70000000000000029e-151

   1. Initial program 85.8%

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

      1. +-commutative 85.8%

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

      2. unpow2 85.8%

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

      3. unpow2 85.8%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 44.5%

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

   5. Taylor expanded in kx around 0 18.2%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\sin ky \cdot kx\right) + \frac{\sin ky}{kx}\right)} \cdot \sin th \]

   6. Taylor expanded in ky around 0 22.6%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot kx + \frac{1}{kx}\right) \cdot \left(\sin th \cdot ky\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -180:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.7 \cdot 10^{-151}:\\ \;\;\;\;\left(kx \cdot 0.16666666666666666 + \frac{1}{kx}\right) \cdot \left(\sin th \cdot ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 32.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-248}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.65e-49)
   (sin th)
   (if (<= ky 4.6e-248) (* ky (/ th (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = sin(th);
	} else if (ky <= 4.6e-248) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.65d-49)) then
        tmp = sin(th)
    else if (ky <= 4.6d-248) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = Math.sin(th);
	} else if (ky <= 4.6e-248) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.65e-49:
		tmp = math.sin(th)
	elif ky <= 4.6e-248:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.65e-49)
		tmp = sin(th);
	elseif (ky <= 4.6e-248)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.65e-49)
		tmp = sin(th);
	elseif (ky <= 4.6e-248)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.65e-49], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 4.6e-248], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.65e-49 or 4.6e-248 < ky

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. unpow2 96.7%

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

      3. unpow2 96.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 30.6%

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

   if -1.65e-49 < ky < 4.6e-248

   1. Initial program 86.9%

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

      1. associate-*l/ 80.3%

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

      2. +-commutative 80.3%

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

      3. unpow2 80.3%

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

      4. unpow2 80.3%

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

      5. hypot-def 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in th around 0 31.0%

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

   5. Taylor expanded in ky around 0 22.0%

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

      1. associate-/l* 23.9%

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

   7. Simplified 23.9%

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

   8. Taylor expanded in ky around 0 22.0%

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

      1. associate-*r/ 23.8%

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

   10. Simplified 23.8%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-248}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 32.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-248}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.65e-49)
   (sin th)
   (if (<= ky 4.6e-248) (* th (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = sin(th);
	} else if (ky <= 4.6e-248) {
		tmp = th * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.65d-49)) then
        tmp = sin(th)
    else if (ky <= 4.6d-248) then
        tmp = th * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = Math.sin(th);
	} else if (ky <= 4.6e-248) {
		tmp = th * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.65e-49:
		tmp = math.sin(th)
	elif ky <= 4.6e-248:
		tmp = th * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.65e-49)
		tmp = sin(th);
	elseif (ky <= 4.6e-248)
		tmp = Float64(th * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.65e-49)
		tmp = sin(th);
	elseif (ky <= 4.6e-248)
		tmp = th * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.65e-49], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 4.6e-248], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.65e-49 or 4.6e-248 < ky

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. unpow2 96.7%

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

      3. unpow2 96.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 30.6%

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

   if -1.65e-49 < ky < 4.6e-248

   1. Initial program 86.9%

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

      1. associate-*l/ 80.3%

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

      2. +-commutative 80.3%

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

      3. unpow2 80.3%

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

      4. unpow2 80.3%

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

      5. hypot-def 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in th around 0 31.0%

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

   5. Taylor expanded in ky around 0 22.0%

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

      1. associate-/l* 23.9%

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

   7. Simplified 23.9%

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

      1. associate-/r/ 24.0%

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

   9. Applied egg-rr 24.0%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-248}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 31.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.4 \cdot 10^{-248}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -4.2e-19)
   (sin th)
   (if (<= ky 3.4e-248) (/ th (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -4.2e-19) {
		tmp = sin(th);
	} else if (ky <= 3.4e-248) {
		tmp = th / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-4.2d-19)) then
        tmp = sin(th)
    else if (ky <= 3.4d-248) then
        tmp = th / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -4.2e-19) {
		tmp = Math.sin(th);
	} else if (ky <= 3.4e-248) {
		tmp = th / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -4.2e-19:
		tmp = math.sin(th)
	elif ky <= 3.4e-248:
		tmp = th / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -4.2e-19)
		tmp = sin(th);
	elseif (ky <= 3.4e-248)
		tmp = Float64(th / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -4.2e-19)
		tmp = sin(th);
	elseif (ky <= 3.4e-248)
		tmp = th / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -4.2e-19], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.4e-248], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -4.1999999999999998e-19 or 3.3999999999999998e-248 < ky

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 31.2%

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

   if -4.1999999999999998e-19 < ky < 3.3999999999999998e-248

   1. Initial program 87.4%

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

      1. associate-*l/ 80.9%

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

      2. +-commutative 80.9%

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

      3. unpow2 80.9%

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

      4. unpow2 80.9%

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

      5. hypot-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in th around 0 31.1%

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

   5. Taylor expanded in ky around 0 22.5%

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

      1. associate-/l* 24.3%

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

   7. Simplified 24.3%

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

   8. Taylor expanded in kx around 0 17.4%

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

      1. associate-/l* 19.5%

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

   10. Simplified 19.5%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.4 \cdot 10^{-248}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 21.9% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.06 \cdot 10^{-81}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.65e-49) th (if (<= ky 1.06e-81) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = th;
	} else if (ky <= 1.06e-81) {
		tmp = th * (ky / 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 <= (-1.65d-49)) then
        tmp = th
    else if (ky <= 1.06d-81) then
        tmp = th * (ky / 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 <= -1.65e-49) {
		tmp = th;
	} else if (ky <= 1.06e-81) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.65e-49:
		tmp = th
	elif ky <= 1.06e-81:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.65e-49)
		tmp = th;
	elseif (ky <= 1.06e-81)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.65e-49)
		tmp = th;
	elseif (ky <= 1.06e-81)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.65e-49], th, If[LessEqual[ky, 1.06e-81], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.65e-49 or 1.05999999999999991e-81 < ky

   1. Initial program 99.5%

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

      1. associate-*l/ 98.8%

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

      2. +-commutative 98.8%

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

      3. unpow2 98.8%

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

      4. unpow2 98.8%

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

      5. hypot-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in th around 0 53.6%

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

   5. Taylor expanded in kx around 0 19.9%

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

   if -1.65e-49 < ky < 1.05999999999999991e-81

   1. Initial program 86.6%

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

      1. associate-*l/ 81.2%

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

      2. +-commutative 81.2%

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

      3. unpow2 81.2%

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

      4. unpow2 81.2%

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

      5. hypot-def 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in th around 0 32.5%

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

   5. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 23.6%

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

   7. Simplified 23.6%

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

   8. Taylor expanded in kx around 0 18.8%

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

      1. *-commutative 18.8%

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

      2. associate-/l* 20.1%

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

   10. Simplified 20.1%

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

       1. associate-/r/ 20.2%

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

   12. Applied egg-rr 20.2%

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

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.06 \cdot 10^{-81}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 18: 21.9% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.65e-49) th (if (<= ky 1.9e-79) (/ th (/ kx ky)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = th;
	} else if (ky <= 1.9e-79) {
		tmp = th / (kx / ky);
	} 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 <= (-1.65d-49)) then
        tmp = th
    else if (ky <= 1.9d-79) then
        tmp = th / (kx / ky)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.65e-49) {
		tmp = th;
	} else if (ky <= 1.9e-79) {
		tmp = th / (kx / ky);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.65e-49:
		tmp = th
	elif ky <= 1.9e-79:
		tmp = th / (kx / ky)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.65e-49)
		tmp = th;
	elseif (ky <= 1.9e-79)
		tmp = Float64(th / Float64(kx / ky));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.65e-49)
		tmp = th;
	elseif (ky <= 1.9e-79)
		tmp = th / (kx / ky);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.65e-49], th, If[LessEqual[ky, 1.9e-79], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.65e-49 or 1.9000000000000001e-79 < ky

   1. Initial program 99.5%

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

      1. associate-*l/ 98.8%

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

      2. +-commutative 98.8%

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

      3. unpow2 98.8%

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

      4. unpow2 98.8%

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

      5. hypot-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in th around 0 53.6%

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

   5. Taylor expanded in kx around 0 19.9%

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

   if -1.65e-49 < ky < 1.9000000000000001e-79

   1. Initial program 86.6%

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

      1. associate-*l/ 81.2%

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

      2. +-commutative 81.2%

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

      3. unpow2 81.2%

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

      4. unpow2 81.2%

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

      5. hypot-def 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in th around 0 32.5%

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

   5. Taylor expanded in ky around 0 22.3%

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

      1. associate-/l* 23.6%

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

   7. Simplified 23.6%

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

   8. Taylor expanded in kx around 0 18.8%

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

      1. associate-/l* 20.3%

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

   10. Simplified 20.3%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 19: 13.9% 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 93.4%

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

   1. associate-*l/ 90.4%

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

   2. +-commutative 90.4%

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

   3. unpow2 90.4%

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

   4. unpow2 90.4%

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

   5. hypot-def 94.8%

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

3. Simplified 94.8%

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

4. Taylor expanded in th around 0 43.6%

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

5. Taylor expanded in kx around 0 13.2%

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

6. Final simplification 13.2%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023278 
(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)))