Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.6%

Time: 23.1s

Alternatives: 17

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: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% 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 94.6%

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

   1. associate-*l/ 91.6%

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

   2. *-commutative 91.6%

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

   3. associate-*l/ 94.6%

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

   4. +-commutative 94.6%

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

   5. unpow2 94.6%

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

   6. unpow2 94.6%

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

   7. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-52}:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \left|\frac{1}{\sin ky}\right|\right)\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-52)
   (* (sin ky) (* th (fabs (/ 1.0 (sin ky)))))
   (if (<= (sin ky) 2e-301)
     (* ky (/ (sin th) (sin kx)))
     (if (<= (sin ky) 4e-8)
       (* (sin th) (fabs (/ (sin ky) (sin kx))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-52) {
		tmp = sin(ky) * (th * fabs((1.0 / sin(ky))));
	} else if (sin(ky) <= 2e-301) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (sin(ky) <= 4e-8) {
		tmp = sin(th) * fabs((sin(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) <= (-5d-52)) then
        tmp = sin(ky) * (th * abs((1.0d0 / sin(ky))))
    else if (sin(ky) <= 2d-301) then
        tmp = ky * (sin(th) / sin(kx))
    else if (sin(ky) <= 4d-8) then
        tmp = sin(th) * abs((sin(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) <= -5e-52) {
		tmp = Math.sin(ky) * (th * Math.abs((1.0 / Math.sin(ky))));
	} else if (Math.sin(ky) <= 2e-301) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 4e-8) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-52:
		tmp = math.sin(ky) * (th * math.fabs((1.0 / math.sin(ky))))
	elif math.sin(ky) <= 2e-301:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 4e-8:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-52)
		tmp = Float64(sin(ky) * Float64(th * abs(Float64(1.0 / sin(ky)))));
	elseif (sin(ky) <= 2e-301)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 4e-8)
		tmp = Float64(sin(th) * abs(Float64(sin(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) <= -5e-52)
		tmp = sin(ky) * (th * abs((1.0 / sin(ky))));
	elseif (sin(ky) <= 2e-301)
		tmp = ky * (sin(th) / sin(kx));
	elseif (sin(ky) <= 4e-8)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-52], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Abs[N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-301], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-8], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -5e-52

   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/ 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 43.6%

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

      1. *-commutative 43.6%

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

      2. +-commutative 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in kx around 0 3.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 31.2%

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

      3. inv-pow 31.2%

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

      4. inv-pow 31.2%

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

      5. pow-prod-up 31.2%

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

      6. metadata-eval 31.2%

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

   9. Applied egg-rr 31.2%

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

       1. metadata-eval 31.2%

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

       2. pow-sqr 31.2%

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

       3. unpow-1 31.2%

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

       4. unpow-1 31.2%

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

       5. rem-sqrt-square 31.2%

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

   11. Simplified 31.2%

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

   if -5e-52 < (sin.f64 ky) < 2.00000000000000013e-301

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. hypot-udef 83.7%

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

      5. unpow2 83.7%

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

      6. unpow2 83.7%

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

      7. +-commutative 83.7%

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

      8. unpow2 83.7%

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

      9. unpow2 83.7%

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

      10. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.7%

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

      3. distribute-neg-frac 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0 61.1%

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

      1. *-lft-identity 61.1%

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

      2. times-frac 64.7%

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

      3. /-rgt-identity 64.7%

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

   10. Simplified 64.7%

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

   if 2.00000000000000013e-301 < (sin.f64 ky) < 4.0000000000000001e-8

   1. Initial program 93.1%

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

   2. Taylor expanded in ky around 0 44.2%

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

      1. add-sqr-sqrt 34.6%

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

      2. sqrt-unprod 57.4%

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

      3. pow2 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. unpow2 57.4%

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

      2. rem-sqrt-square 73.2%

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

   6. Simplified 73.2%

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

   if 4.0000000000000001e-8 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 63.3%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-52}:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \left|\frac{1}{\sin ky}\right|\right)\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 50.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.019:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.019)
   (fabs (* (sin ky) (/ th (sin ky))))
   (if (<= (sin ky) 2e-301)
     (* ky (/ (sin th) (sin kx)))
     (if (<= (sin ky) 4e-8) (* (sin th) (fabs (/ ky (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.019) {
		tmp = fabs((sin(ky) * (th / sin(ky))));
	} else if (sin(ky) <= 2e-301) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (sin(ky) <= 4e-8) {
		tmp = sin(th) * fabs((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) <= (-0.019d0)) then
        tmp = abs((sin(ky) * (th / sin(ky))))
    else if (sin(ky) <= 2d-301) then
        tmp = ky * (sin(th) / sin(kx))
    else if (sin(ky) <= 4d-8) then
        tmp = sin(th) * abs((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) <= -0.019) {
		tmp = Math.abs((Math.sin(ky) * (th / Math.sin(ky))));
	} else if (Math.sin(ky) <= 2e-301) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 4e-8) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.019:
		tmp = math.fabs((math.sin(ky) * (th / math.sin(ky))))
	elif math.sin(ky) <= 2e-301:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 4e-8:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.019)
		tmp = abs(Float64(sin(ky) * Float64(th / sin(ky))));
	elseif (sin(ky) <= 2e-301)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 4e-8)
		tmp = Float64(sin(th) * abs(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) <= -0.019)
		tmp = abs((sin(ky) * (th / sin(ky))));
	elseif (sin(ky) <= 2e-301)
		tmp = ky * (sin(th) / sin(kx));
	elseif (sin(ky) <= 4e-8)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.019], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-301], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-8], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.0189999999999999995

   1. Initial program 99.7%

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

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 41.6%

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

      1. *-commutative 41.6%

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

      2. +-commutative 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 17.2%

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

      3. pow2 17.2%

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

      4. *-commutative 17.2%

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

      5. associate-*l/ 17.3%

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

      6. *-un-lft-identity 17.3%

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

   9. Applied egg-rr 17.3%

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

       1. unpow2 17.3%

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

       2. rem-sqrt-square 23.2%

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

   11. Simplified 23.2%

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

   if -0.0189999999999999995 < (sin.f64 ky) < 2.00000000000000013e-301

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. unpow2 86.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.7%

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

      4. hypot-udef 86.2%

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

      5. unpow2 86.2%

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

      6. unpow2 86.2%

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

      7. +-commutative 86.2%

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

      8. unpow2 86.2%

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

      9. unpow2 86.2%

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

      10. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.6%

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

      3. distribute-neg-frac 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in ky around 0 55.4%

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

      1. *-lft-identity 55.4%

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

      2. times-frac 58.2%

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

      3. /-rgt-identity 58.2%

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

   10. Simplified 58.2%

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

   if 2.00000000000000013e-301 < (sin.f64 ky) < 4.0000000000000001e-8

   1. Initial program 93.1%

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

   2. Taylor expanded in ky around 0 44.2%

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

      1. add-sqr-sqrt 34.6%

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

      2. sqrt-unprod 57.4%

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

      3. pow2 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. unpow2 57.4%

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

      2. rem-sqrt-square 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in ky around 0 73.2%

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

   if 4.0000000000000001e-8 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 63.3%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.019:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 50.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.07:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.07)
   (fabs (* (sin ky) (/ th (sin ky))))
   (if (<= (sin ky) -1e-297)
     (* (sin th) (/ (sin ky) (sin kx)))
     (if (<= (sin ky) 4e-8) (* (sin th) (fabs (/ ky (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.07) {
		tmp = fabs((sin(ky) * (th / sin(ky))));
	} else if (sin(ky) <= -1e-297) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (sin(ky) <= 4e-8) {
		tmp = sin(th) * fabs((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) <= (-0.07d0)) then
        tmp = abs((sin(ky) * (th / sin(ky))))
    else if (sin(ky) <= (-1d-297)) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else if (sin(ky) <= 4d-8) then
        tmp = sin(th) * abs((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) <= -0.07) {
		tmp = Math.abs((Math.sin(ky) * (th / Math.sin(ky))));
	} else if (Math.sin(ky) <= -1e-297) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else if (Math.sin(ky) <= 4e-8) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.07:
		tmp = math.fabs((math.sin(ky) * (th / math.sin(ky))))
	elif math.sin(ky) <= -1e-297:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	elif math.sin(ky) <= 4e-8:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.07)
		tmp = abs(Float64(sin(ky) * Float64(th / sin(ky))));
	elseif (sin(ky) <= -1e-297)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (sin(ky) <= 4e-8)
		tmp = Float64(sin(th) * abs(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) <= -0.07)
		tmp = abs((sin(ky) * (th / sin(ky))));
	elseif (sin(ky) <= -1e-297)
		tmp = sin(th) * (sin(ky) / sin(kx));
	elseif (sin(ky) <= 4e-8)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.07], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-297], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-8], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.070000000000000007

   1. Initial program 99.7%

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

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 43.3%

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

      1. *-commutative 43.3%

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

      2. +-commutative 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 17.9%

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

      3. pow2 17.9%

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

      4. *-commutative 17.9%

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

      5. associate-*l/ 17.9%

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

      6. *-un-lft-identity 17.9%

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

   9. Applied egg-rr 17.9%

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

       1. unpow2 17.9%

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

       2. rem-sqrt-square 24.1%

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

   11. Simplified 24.1%

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

   if -0.070000000000000007 < (sin.f64 ky) < -1.00000000000000004e-297

   1. Initial program 87.9%

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

   2. Taylor expanded in ky around 0 54.4%

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

   if -1.00000000000000004e-297 < (sin.f64 ky) < 4.0000000000000001e-8

   1. Initial program 90.9%

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

   2. Taylor expanded in ky around 0 48.2%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-unprod 58.1%

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

      3. pow2 58.1%

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

   4. Applied egg-rr 58.1%

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

      1. unpow2 58.1%

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

      2. rem-sqrt-square 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in ky around 0 76.1%

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

   if 4.0000000000000001e-8 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 63.3%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.07:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 50.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.07:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.07)
   (fabs (* (sin ky) (/ th (sin ky))))
   (if (<= (sin ky) 2e-301)
     (* (sin ky) (/ (sin th) (sin kx)))
     (if (<= (sin ky) 4e-8) (* (sin th) (fabs (/ ky (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.07) {
		tmp = fabs((sin(ky) * (th / sin(ky))));
	} else if (sin(ky) <= 2e-301) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(ky) <= 4e-8) {
		tmp = sin(th) * fabs((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) <= (-0.07d0)) then
        tmp = abs((sin(ky) * (th / sin(ky))))
    else if (sin(ky) <= 2d-301) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else if (sin(ky) <= 4d-8) then
        tmp = sin(th) * abs((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) <= -0.07) {
		tmp = Math.abs((Math.sin(ky) * (th / Math.sin(ky))));
	} else if (Math.sin(ky) <= 2e-301) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 4e-8) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.07:
		tmp = math.fabs((math.sin(ky) * (th / math.sin(ky))))
	elif math.sin(ky) <= 2e-301:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 4e-8:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.07)
		tmp = abs(Float64(sin(ky) * Float64(th / sin(ky))));
	elseif (sin(ky) <= 2e-301)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 4e-8)
		tmp = Float64(sin(th) * abs(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) <= -0.07)
		tmp = abs((sin(ky) * (th / sin(ky))));
	elseif (sin(ky) <= 2e-301)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(ky) <= 4e-8)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.07], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-301], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-8], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.070000000000000007

   1. Initial program 99.7%

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

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 43.3%

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

      1. *-commutative 43.3%

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

      2. +-commutative 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 17.9%

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

      3. pow2 17.9%

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

      4. *-commutative 17.9%

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

      5. associate-*l/ 17.9%

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

      6. *-un-lft-identity 17.9%

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

   9. Applied egg-rr 17.9%

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

       1. unpow2 17.9%

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

       2. rem-sqrt-square 24.1%

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

   11. Simplified 24.1%

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

   if -0.070000000000000007 < (sin.f64 ky) < 2.00000000000000013e-301

   1. Initial program 86.7%

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

      1. associate-*l/ 79.4%

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

      2. *-commutative 79.4%

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

      3. associate-*l/ 86.8%

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

      4. +-commutative 86.8%

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

      5. unpow2 86.8%

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

      6. unpow2 86.8%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 56.6%

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

   if 2.00000000000000013e-301 < (sin.f64 ky) < 4.0000000000000001e-8

   1. Initial program 93.1%

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

   2. Taylor expanded in ky around 0 44.2%

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

      1. add-sqr-sqrt 34.6%

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

      2. sqrt-unprod 57.4%

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

      3. pow2 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. unpow2 57.4%

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

      2. rem-sqrt-square 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in ky around 0 73.2%

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

   if 4.0000000000000001e-8 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 63.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.07:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 54.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-52}:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \left|\frac{1}{\sin ky}\right|\right)\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-52)
   (* (sin ky) (* th (fabs (/ 1.0 (sin ky)))))
   (if (<= (sin ky) 2e-301)
     (* ky (/ (sin th) (sin kx)))
     (if (<= (sin ky) 4e-8) (* (sin th) (fabs (/ ky (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-52) {
		tmp = sin(ky) * (th * fabs((1.0 / sin(ky))));
	} else if (sin(ky) <= 2e-301) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (sin(ky) <= 4e-8) {
		tmp = sin(th) * fabs((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) <= (-5d-52)) then
        tmp = sin(ky) * (th * abs((1.0d0 / sin(ky))))
    else if (sin(ky) <= 2d-301) then
        tmp = ky * (sin(th) / sin(kx))
    else if (sin(ky) <= 4d-8) then
        tmp = sin(th) * abs((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) <= -5e-52) {
		tmp = Math.sin(ky) * (th * Math.abs((1.0 / Math.sin(ky))));
	} else if (Math.sin(ky) <= 2e-301) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= 4e-8) {
		tmp = Math.sin(th) * Math.abs((ky / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-52:
		tmp = math.sin(ky) * (th * math.fabs((1.0 / math.sin(ky))))
	elif math.sin(ky) <= 2e-301:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= 4e-8:
		tmp = math.sin(th) * math.fabs((ky / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-52)
		tmp = Float64(sin(ky) * Float64(th * abs(Float64(1.0 / sin(ky)))));
	elseif (sin(ky) <= 2e-301)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 4e-8)
		tmp = Float64(sin(th) * abs(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) <= -5e-52)
		tmp = sin(ky) * (th * abs((1.0 / sin(ky))));
	elseif (sin(ky) <= 2e-301)
		tmp = ky * (sin(th) / sin(kx));
	elseif (sin(ky) <= 4e-8)
		tmp = sin(th) * abs((ky / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-52], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Abs[N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-301], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-8], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -5e-52

   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/ 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 43.6%

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

      1. *-commutative 43.6%

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

      2. +-commutative 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in kx around 0 3.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 31.2%

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

      3. inv-pow 31.2%

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

      4. inv-pow 31.2%

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

      5. pow-prod-up 31.2%

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

      6. metadata-eval 31.2%

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

   9. Applied egg-rr 31.2%

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

       1. metadata-eval 31.2%

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

       2. pow-sqr 31.2%

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

       3. unpow-1 31.2%

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

       4. unpow-1 31.2%

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

       5. rem-sqrt-square 31.2%

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

   11. Simplified 31.2%

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

   if -5e-52 < (sin.f64 ky) < 2.00000000000000013e-301

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. hypot-udef 83.7%

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

      5. unpow2 83.7%

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

      6. unpow2 83.7%

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

      7. +-commutative 83.7%

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

      8. unpow2 83.7%

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

      9. unpow2 83.7%

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

      10. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.7%

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

      3. distribute-neg-frac 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0 61.1%

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

      1. *-lft-identity 61.1%

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

      2. times-frac 64.7%

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

      3. /-rgt-identity 64.7%

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

   10. Simplified 64.7%

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

   if 2.00000000000000013e-301 < (sin.f64 ky) < 4.0000000000000001e-8

   1. Initial program 93.1%

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

   2. Taylor expanded in ky around 0 44.2%

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

      1. add-sqr-sqrt 34.6%

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

      2. sqrt-unprod 57.4%

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

      3. pow2 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. unpow2 57.4%

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

      2. rem-sqrt-square 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in ky around 0 73.2%

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

   if 4.0000000000000001e-8 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 63.3%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-52}:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \left|\frac{1}{\sin ky}\right|\right)\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-301}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 66.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.002)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin th) 1e-15)
     (* th (/ (sin ky) (hypot (sin kx) (sin ky))))
     (/ (sin ky) (fabs (/ (sin kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.002) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(th) <= 1e-15) {
		tmp = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	} else {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.002) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(th) <= 1e-15) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.002:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(th) <= 1e-15:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.002)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(th) <= 1e-15)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.002)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(th) <= 1e-15)
		tmp = th * (sin(ky) / hypot(sin(kx), sin(ky)));
	else
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.002], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-15], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -2e-3

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*l/ 92.4%

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

      4. +-commutative 92.4%

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

      5. unpow2 92.4%

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

      6. unpow2 92.4%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 21.0%

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

   if -2e-3 < (sin.f64 th) < 1.0000000000000001e-15

   1. Initial program 96.0%

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

      1. associate-*l/ 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*l/ 96.0%

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

      4. +-commutative 96.0%

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

      5. unpow2 96.0%

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

      6. unpow2 96.0%

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

      7. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 95.4%

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

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. expm1-log1p-u 95.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot th\right) \cdot \sin ky\right)\right)} \]

      2. expm1-udef 25.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot th\right) \cdot \sin ky\right)} - 1} \]

   8. Applied egg-rr 25.5%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   10. Simplified 99.7%

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

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

   1. Initial program 95.0%

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

   2. Taylor expanded in ky around 0 23.0%

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

      1. associate-*l/ 23.1%

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

      2. associate-/l* 23.0%

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

   4. Applied egg-rr 23.0%

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

      1. add-sqr-sqrt 21.8%

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

      2. sqrt-unprod 38.9%

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

      3. pow2 38.9%

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

   6. Applied egg-rr 38.9%

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

      1. unpow2 38.9%

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

      2. rem-sqrt-square 39.5%

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

   8. Simplified 39.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 8: 66.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.002)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin th) 1e-15)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (/ (sin ky) (fabs (/ (sin kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.002) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(th) <= 1e-15) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.002) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(th) <= 1e-15) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.002:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(th) <= 1e-15:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.002)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(th) <= 1e-15)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.002)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(th) <= 1e-15)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.002], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-15], 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[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -2e-3

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*l/ 92.4%

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

      4. +-commutative 92.4%

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

      5. unpow2 92.4%

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

      6. unpow2 92.4%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 21.0%

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

   if -2e-3 < (sin.f64 th) < 1.0000000000000001e-15

   1. Initial program 96.0%

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

      1. associate-*l/ 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*l/ 96.0%

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

      4. +-commutative 96.0%

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

      5. unpow2 96.0%

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

      6. unpow2 96.0%

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

      7. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 95.4%

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

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-log-exp 25.4%

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

      2. *-commutative 25.4%

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

      3. exp-prod 25.4%

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

      4. *-commutative 25.4%

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

      5. sqrt-div 25.4%

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

      6. metadata-eval 25.4%

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

      7. +-commutative 25.4%

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

      8. unpow2 25.4%

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

      9. unpow2 25.4%

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

      10. hypot-udef 25.4%

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

      11. un-div-inv 25.4%

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

   8. Applied egg-rr 25.4%

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

      1. log-pow 68.8%

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

      2. hypot-def 68.5%

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

      3. unpow2 68.5%

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

      4. unpow2 68.5%

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

      5. +-commutative 68.5%

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

      6. unpow2 68.5%

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

      7. unpow2 68.5%

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

      8. hypot-def 68.8%

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

      9. rem-log-exp 99.7%

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

   10. Simplified 99.7%

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

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

   1. Initial program 95.0%

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

   2. Taylor expanded in ky around 0 23.0%

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

      1. associate-*l/ 23.1%

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

      2. associate-/l* 23.0%

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

   4. Applied egg-rr 23.0%

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

      1. add-sqr-sqrt 21.8%

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

      2. sqrt-unprod 38.9%

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

      3. pow2 38.9%

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

   6. Applied egg-rr 38.9%

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

      1. unpow2 38.9%

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

      2. rem-sqrt-square 39.5%

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

   8. Simplified 39.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 9: 66.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.002)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin th) 1e-15)
     (/ th (/ (hypot (sin ky) (sin kx)) (sin ky)))
     (/ (sin ky) (fabs (/ (sin kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.002) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(th) <= 1e-15) {
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	} else {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.002) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(th) <= 1e-15) {
		tmp = th / (Math.hypot(Math.sin(ky), Math.sin(kx)) / Math.sin(ky));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.002:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(th) <= 1e-15:
		tmp = th / (math.hypot(math.sin(ky), math.sin(kx)) / math.sin(ky))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.002)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(th) <= 1e-15)
		tmp = Float64(th / Float64(hypot(sin(ky), sin(kx)) / sin(ky)));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.002)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(th) <= 1e-15)
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	else
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.002], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-15], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -2e-3

   1. Initial program 92.3%

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

      1. associate-*l/ 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*l/ 92.4%

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

      4. +-commutative 92.4%

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

      5. unpow2 92.4%

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

      6. unpow2 92.4%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 21.0%

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

   if -2e-3 < (sin.f64 th) < 1.0000000000000001e-15

   1. Initial program 96.0%

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

      1. associate-*l/ 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*l/ 96.0%

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

      4. +-commutative 96.0%

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

      5. unpow2 96.0%

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

      6. unpow2 96.0%

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

      7. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 95.4%

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

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. expm1-log1p-u 95.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot th\right) \cdot \sin ky\right)\right)} \]

      2. expm1-udef 25.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot th\right) \cdot \sin ky\right)} - 1} \]

   8. Applied egg-rr 25.5%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 89.7%

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

      4. associate-/l* 99.7%

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

      5. hypot-def 96.0%

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

      6. unpow2 96.0%

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

      7. unpow2 96.0%

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

      8. +-commutative 96.0%

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

      9. unpow2 96.0%

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

      10. unpow2 96.0%

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

      11. hypot-def 99.7%

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

   10. Simplified 99.7%

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

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

   1. Initial program 95.0%

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

   2. Taylor expanded in ky around 0 23.0%

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

      1. associate-*l/ 23.1%

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

      2. associate-/l* 23.0%

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

   4. Applied egg-rr 23.0%

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

      1. add-sqr-sqrt 21.8%

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

      2. sqrt-unprod 38.9%

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

      3. pow2 38.9%

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

   6. Applied egg-rr 38.9%

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

      1. unpow2 38.9%

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

      2. rem-sqrt-square 39.5%

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

   8. Simplified 39.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 10: 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 94.6%

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

   1. +-commutative 94.6%

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

   2. unpow2 94.6%

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

   3. unpow2 94.6%

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

   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. Final simplification 99.7%

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

Alternative 11: 44.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.019:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-149}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.019)
   (fabs (* (sin ky) (/ th (sin ky))))
   (if (<= (sin ky) 2e-149) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.019) {
		tmp = fabs((sin(ky) * (th / sin(ky))));
	} else if (sin(ky) <= 2e-149) {
		tmp = ky * (sin(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 (sin(ky) <= (-0.019d0)) then
        tmp = abs((sin(ky) * (th / sin(ky))))
    else if (sin(ky) <= 2d-149) then
        tmp = ky * (sin(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 (Math.sin(ky) <= -0.019) {
		tmp = Math.abs((Math.sin(ky) * (th / Math.sin(ky))));
	} else if (Math.sin(ky) <= 2e-149) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.019:
		tmp = math.fabs((math.sin(ky) * (th / math.sin(ky))))
	elif math.sin(ky) <= 2e-149:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.019)
		tmp = abs(Float64(sin(ky) * Float64(th / sin(ky))));
	elseif (sin(ky) <= 2e-149)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.019)
		tmp = abs((sin(ky) * (th / sin(ky))));
	elseif (sin(ky) <= 2e-149)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.019], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-149], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0189999999999999995

   1. Initial program 99.7%

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

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 41.6%

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

      1. *-commutative 41.6%

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

      2. +-commutative 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 17.2%

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

      3. pow2 17.2%

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

      4. *-commutative 17.2%

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

      5. associate-*l/ 17.3%

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

      6. *-un-lft-identity 17.3%

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

   9. Applied egg-rr 17.3%

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

       1. unpow2 17.3%

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

       2. rem-sqrt-square 23.2%

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

   11. Simplified 23.2%

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

   if -0.0189999999999999995 < (sin.f64 ky) < 1.99999999999999996e-149

   1. Initial program 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. unpow2 86.0%

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

      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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

      4. hypot-udef 86.0%

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

      5. unpow2 86.0%

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

      6. unpow2 86.0%

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

      7. +-commutative 86.0%

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

      8. unpow2 86.0%

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

      9. unpow2 86.0%

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

      10. hypot-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

      3. distribute-neg-frac 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in ky around 0 58.7%

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

      1. *-lft-identity 58.7%

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

      2. times-frac 60.8%

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

      3. /-rgt-identity 60.8%

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

   10. Simplified 60.8%

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

   if 1.99999999999999996e-149 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 57.7%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.019:\\ \;\;\;\;\left|\sin ky \cdot \frac{th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-149}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 40.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-149}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-149) (* ky (/ (sin th) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-149) {
		tmp = ky * (sin(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 (sin(ky) <= 2d-149) then
        tmp = ky * (sin(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 (Math.sin(ky) <= 2e-149) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-149:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-149)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-149)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-149], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 1.99999999999999996e-149

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. unpow2 91.6%

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

      3. unpow2 91.6%

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

      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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.7%

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

      4. hypot-udef 91.6%

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

      5. unpow2 91.6%

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

      6. unpow2 91.6%

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

      7. +-commutative 91.6%

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

      8. unpow2 91.6%

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

      9. unpow2 91.6%

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

      10. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.6%

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

      3. distribute-neg-frac 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in ky around 0 35.6%

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

      1. *-lft-identity 35.6%

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

      2. times-frac 36.8%

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

      3. /-rgt-identity 36.8%

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

   10. Simplified 36.8%

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

   if 1.99999999999999996e-149 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 57.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-149}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 32.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -0.018 \lor \neg \left(ky \leq 3.7 \cdot 10^{-149}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -0.018) (not (<= ky 3.7e-149)))
   (sin th)
   (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.018) || !(ky <= 3.7e-149)) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-0.018d0)) .or. (.not. (ky <= 3.7d-149))) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.018) || !(ky <= 3.7e-149)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -0.018) or not (ky <= 3.7e-149):
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -0.018) || !(ky <= 3.7e-149))
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -0.018) || ~((ky <= 3.7e-149)))
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -0.018], N[Not[LessEqual[ky, 3.7e-149]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -0.0179999999999999986 or 3.6999999999999999e-149 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 34.7%

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

   if -0.0179999999999999986 < ky < 3.6999999999999999e-149

   1. Initial program 85.7%

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

      1. associate-*l/ 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*l/ 85.7%

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

      4. +-commutative 85.7%

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

      5. unpow2 85.7%

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

      6. unpow2 85.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. +-commutative 50.4%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in ky around 0 37.6%

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

      1. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

   10. Taylor expanded in ky around 0 37.6%

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

       1. associate-*r/ 39.6%

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

   12. Simplified 39.6%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.018 \lor \neg \left(ky \leq 3.7 \cdot 10^{-149}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]

Alternative 14: 33.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -0.018 \lor \neg \left(ky \leq 5.6 \cdot 10^{-149}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -0.018) (not (<= ky 5.6e-149)))
   (sin th)
   (* th (/ ky (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.018) || !(ky <= 5.6e-149)) {
		tmp = sin(th);
	} else {
		tmp = th * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-0.018d0)) .or. (.not. (ky <= 5.6d-149))) then
        tmp = sin(th)
    else
        tmp = th * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -0.018) || !(ky <= 5.6e-149)) {
		tmp = Math.sin(th);
	} else {
		tmp = th * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -0.018) or not (ky <= 5.6e-149):
		tmp = math.sin(th)
	else:
		tmp = th * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -0.018) || !(ky <= 5.6e-149))
		tmp = sin(th);
	else
		tmp = Float64(th * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -0.018) || ~((ky <= 5.6e-149)))
		tmp = sin(th);
	else
		tmp = th * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -0.018], N[Not[LessEqual[ky, 5.6e-149]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -0.0179999999999999986 or 5.5999999999999997e-149 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 34.7%

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

   if -0.0179999999999999986 < ky < 5.5999999999999997e-149

   1. Initial program 85.7%

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

      1. associate-*l/ 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*l/ 85.7%

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

      4. +-commutative 85.7%

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

      5. unpow2 85.7%

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

      6. unpow2 85.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. +-commutative 50.4%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in ky around 0 37.6%

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

      1. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

       1. associate-/r/ 39.7%

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

   11. Applied egg-rr 39.7%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -0.018 \lor \neg \left(ky \leq 5.6 \cdot 10^{-149}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 15: 31.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.56 \cdot 10^{-19} \lor \neg \left(ky \leq 2.1 \cdot 10^{-149}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -1.56e-19) (not (<= ky 2.1e-149))) (sin th) (* th (/ ky kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.56e-19) || !(ky <= 2.1e-149)) {
		tmp = sin(th);
	} else {
		tmp = th * (ky / kx);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-1.56d-19)) .or. (.not. (ky <= 2.1d-149))) then
        tmp = sin(th)
    else
        tmp = th * (ky / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.56e-19) || !(ky <= 2.1e-149)) {
		tmp = Math.sin(th);
	} else {
		tmp = th * (ky / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -1.56e-19) or not (ky <= 2.1e-149):
		tmp = math.sin(th)
	else:
		tmp = th * (ky / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -1.56e-19) || !(ky <= 2.1e-149))
		tmp = sin(th);
	else
		tmp = Float64(th * Float64(ky / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -1.56e-19) || ~((ky <= 2.1e-149)))
		tmp = sin(th);
	else
		tmp = th * (ky / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -1.56e-19], N[Not[LessEqual[ky, 2.1e-149]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.56000000000000003e-19 or 2.10000000000000011e-149 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   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 34.3%

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

   if -1.56000000000000003e-19 < ky < 2.10000000000000011e-149

   1. Initial program 85.4%

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

      1. associate-*l/ 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-*l/ 85.4%

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

      4. +-commutative 85.4%

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

      5. unpow2 85.4%

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

      6. unpow2 85.4%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. +-commutative 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in ky around 0 38.3%

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

      1. associate-/l* 40.4%

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

   9. Simplified 40.4%

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

   10. Taylor expanded in kx around 0 34.8%

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

       1. associate-/r/ 34.9%

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

   12. Applied egg-rr 34.9%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.56 \cdot 10^{-19} \lor \neg \left(ky \leq 2.1 \cdot 10^{-149}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \end{array} \]

Alternative 16: 21.8% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -7 \cdot 10^{-20}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 9.5 \cdot 10^{-89}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -7e-20) th (if (<= ky 9.5e-89) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -7e-20) {
		tmp = th;
	} else if (ky <= 9.5e-89) {
		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 <= (-7d-20)) then
        tmp = th
    else if (ky <= 9.5d-89) 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 <= -7e-20) {
		tmp = th;
	} else if (ky <= 9.5e-89) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -7e-20:
		tmp = th
	elif ky <= 9.5e-89:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -7e-20)
		tmp = th;
	elseif (ky <= 9.5e-89)
		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 <= -7e-20)
		tmp = th;
	elseif (ky <= 9.5e-89)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -7e-20], th, If[LessEqual[ky, 9.5e-89], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -7.00000000000000007e-20 or 9.50000000000000028e-89 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 44.5%

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

      1. *-commutative 44.5%

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

      2. +-commutative 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in kx around 0 16.9%

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

   if -7.00000000000000007e-20 < ky < 9.50000000000000028e-89

   1. Initial program 86.8%

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

      1. associate-*l/ 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*l/ 86.9%

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

      4. +-commutative 86.9%

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

      5. unpow2 86.9%

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

      6. unpow2 86.9%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 49.7%

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

      1. *-commutative 49.7%

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

      2. +-commutative 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in ky around 0 35.7%

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

      1. associate-/l* 37.6%

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

   9. Simplified 37.6%

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

   10. Taylor expanded in kx around 0 32.8%

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

       1. associate-/r/ 32.9%

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

   12. Applied egg-rr 32.9%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -7 \cdot 10^{-20}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 9.5 \cdot 10^{-89}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 17: 13.4% 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 94.6%

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

   1. associate-*l/ 91.6%

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

   2. *-commutative 91.6%

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

   3. associate-*l/ 94.6%

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

   4. +-commutative 94.6%

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

   5. unpow2 94.6%

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

   6. unpow2 94.6%

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

   7. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in th around 0 46.5%

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

   1. *-commutative 46.5%

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

   2. +-commutative 46.5%

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

6. Simplified 46.5%

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

7. Taylor expanded in kx around 0 13.1%

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

8. Final simplification 13.1%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023310 
(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)))