Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 25.3s

Alternatives: 20

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. +-commutative 96.0%

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

   2. unpow2 96.0%

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

   3. unpow2 96.0%

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

   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. Final simplification 99.7%

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

Alternative 2: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \mathsf{log1p}\left(\frac{ky}{\sin kx}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.0001)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) -1e-153)
     (* (sin ky) (/ (sin th) (- kx)))
     (if (<= (sin kx) 2e-44) (sin th) (* (sin th) (log1p (/ ky (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.0001) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= -1e-153) {
		tmp = sin(ky) * (sin(th) / -kx);
	} else if (sin(kx) <= 2e-44) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * log1p((ky / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.0001) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= -1e-153) {
		tmp = Math.sin(ky) * (Math.sin(th) / -kx);
	} else if (Math.sin(kx) <= 2e-44) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * Math.log1p((ky / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.0001:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= -1e-153:
		tmp = math.sin(ky) * (math.sin(th) / -kx)
	elif math.sin(kx) <= 2e-44:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * math.log1p((ky / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.0001)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= -1e-153)
		tmp = Float64(sin(ky) * Float64(sin(th) / Float64(-kx)));
	elseif (sin(kx) <= 2e-44)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * log1p(Float64(ky / sin(kx))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.0001], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -1e-153], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / (-kx)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-44], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[Log[1 + N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 16.0%

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

      1. associate-/l* 16.0%

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

   6. Simplified 16.0%

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

      1. add-sqr-sqrt 13.1%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

      4. div-inv 25.1%

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

      5. clear-num 25.1%

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

   8. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 35.4%

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

      3. associate-*r/ 35.4%

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

      4. associate-*l/ 35.4%

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

      5. *-commutative 35.4%

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

   10. Simplified 35.4%

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

   if -1.00000000000000005e-4 < (sin.f64 kx) < -1.00000000000000004e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 7.2%

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

   5. Taylor expanded in kx around 0 7.3%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 12.7%

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

      3. pow2 12.7%

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

      4. associate-/l* 12.7%

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

   7. Applied egg-rr 12.7%

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

   8. Taylor expanded in kx around -inf 44.0%

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

      1. mul-1-neg 44.0%

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

      2. associate-*l/ 47.9%

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

      3. *-commutative 47.9%

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

      4. distribute-rgt-neg-in 47.9%

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

      5. *-lft-identity 47.9%

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

      6. metadata-eval 47.9%

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

      7. times-frac 47.9%

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

      8. neg-mul-1 47.9%

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

      9. neg-mul-1 47.9%

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

      10. distribute-frac-neg 47.9%

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

      11. remove-double-neg 47.9%

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

   10. Simplified 47.9%

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

   if -1.00000000000000004e-153 < (sin.f64 kx) < 1.99999999999999991e-44

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

      2. unpow2 89.7%

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

      3. unpow2 89.7%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 44.0%

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

   if 1.99999999999999991e-44 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-def 99.4%

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

   3. Simplified 99.4%

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

      1. log1p-expm1-u 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 52.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{ky}{\sin kx}}\right) \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \mathsf{log1p}\left(\frac{ky}{\sin kx}\right)\\ \end{array} \]

Alternative 3: 44.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot t_1\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-150}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.0001)
     (fabs (* ky t_1))
     (if (<= (sin kx) -1e-153)
       (* (sin ky) (/ (sin th) (- kx)))
       (if (<= (sin kx) 1e-150) (sin th) (* (sin ky) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.0001) {
		tmp = fabs((ky * t_1));
	} else if (sin(kx) <= -1e-153) {
		tmp = sin(ky) * (sin(th) / -kx);
	} else if (sin(kx) <= 1e-150) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    if (sin(kx) <= (-0.0001d0)) then
        tmp = abs((ky * t_1))
    else if (sin(kx) <= (-1d-153)) then
        tmp = sin(ky) * (sin(th) / -kx)
    else if (sin(kx) <= 1d-150) then
        tmp = sin(th)
    else
        tmp = sin(ky) * t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.0001) {
		tmp = Math.abs((ky * t_1));
	} else if (Math.sin(kx) <= -1e-153) {
		tmp = Math.sin(ky) * (Math.sin(th) / -kx);
	} else if (Math.sin(kx) <= 1e-150) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.0001:
		tmp = math.fabs((ky * t_1))
	elif math.sin(kx) <= -1e-153:
		tmp = math.sin(ky) * (math.sin(th) / -kx)
	elif math.sin(kx) <= 1e-150:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.0001)
		tmp = abs(Float64(ky * t_1));
	elseif (sin(kx) <= -1e-153)
		tmp = Float64(sin(ky) * Float64(sin(th) / Float64(-kx)));
	elseif (sin(kx) <= 1e-150)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.0001)
		tmp = abs((ky * t_1));
	elseif (sin(kx) <= -1e-153)
		tmp = sin(ky) * (sin(th) / -kx);
	elseif (sin(kx) <= 1e-150)
		tmp = sin(th);
	else
		tmp = sin(ky) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.0001], N[Abs[N[(ky * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -1e-153], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / (-kx)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-150], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 16.0%

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

      1. associate-/l* 16.0%

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

   6. Simplified 16.0%

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

      1. add-sqr-sqrt 13.1%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

      4. div-inv 25.1%

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

      5. clear-num 25.1%

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

   8. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 35.4%

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

      3. associate-*r/ 35.4%

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

      4. associate-*l/ 35.4%

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

      5. *-commutative 35.4%

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

   10. Simplified 35.4%

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

   if -1.00000000000000005e-4 < (sin.f64 kx) < -1.00000000000000004e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 7.2%

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

   5. Taylor expanded in kx around 0 7.3%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 12.7%

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

      3. pow2 12.7%

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

      4. associate-/l* 12.7%

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

   7. Applied egg-rr 12.7%

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

   8. Taylor expanded in kx around -inf 44.0%

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

      1. mul-1-neg 44.0%

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

      2. associate-*l/ 47.9%

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

      3. *-commutative 47.9%

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

      4. distribute-rgt-neg-in 47.9%

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

      5. *-lft-identity 47.9%

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

      6. metadata-eval 47.9%

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

      7. times-frac 47.9%

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

      8. neg-mul-1 47.9%

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

      9. neg-mul-1 47.9%

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

      10. distribute-frac-neg 47.9%

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

      11. remove-double-neg 47.9%

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

   10. Simplified 47.9%

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

   if -1.00000000000000004e-153 < (sin.f64 kx) < 1.00000000000000001e-150

   1. Initial program 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

      4. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 48.3%

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

   if 1.00000000000000001e-150 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. associate-*l/ 98.4%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. unpow2 99.5%

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

      6. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 53.8%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-150}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 4: 44.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-150}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.0001)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) -1e-153)
     (* (sin ky) (/ (sin th) (- kx)))
     (if (<= (sin kx) 1e-150) (sin th) (/ (sin ky) (/ (sin kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.0001) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= -1e-153) {
		tmp = sin(ky) * (sin(th) / -kx);
	} else if (sin(kx) <= 1e-150) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.0001d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= (-1d-153)) then
        tmp = sin(ky) * (sin(th) / -kx)
    else if (sin(kx) <= 1d-150) then
        tmp = sin(th)
    else
        tmp = sin(ky) / (sin(kx) / sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.0001) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= -1e-153) {
		tmp = Math.sin(ky) * (Math.sin(th) / -kx);
	} else if (Math.sin(kx) <= 1e-150) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.0001:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= -1e-153:
		tmp = math.sin(ky) * (math.sin(th) / -kx)
	elif math.sin(kx) <= 1e-150:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.0001)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= -1e-153)
		tmp = Float64(sin(ky) * Float64(sin(th) / Float64(-kx)));
	elseif (sin(kx) <= 1e-150)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.0001)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= -1e-153)
		tmp = sin(ky) * (sin(th) / -kx);
	elseif (sin(kx) <= 1e-150)
		tmp = sin(th);
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.0001], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -1e-153], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / (-kx)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-150], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 16.0%

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

      1. associate-/l* 16.0%

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

   6. Simplified 16.0%

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

      1. add-sqr-sqrt 13.1%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

      4. div-inv 25.1%

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

      5. clear-num 25.1%

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

   8. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 35.4%

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

      3. associate-*r/ 35.4%

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

      4. associate-*l/ 35.4%

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

      5. *-commutative 35.4%

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

   10. Simplified 35.4%

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

   if -1.00000000000000005e-4 < (sin.f64 kx) < -1.00000000000000004e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 7.2%

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

   5. Taylor expanded in kx around 0 7.3%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 12.7%

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

      3. pow2 12.7%

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

      4. associate-/l* 12.7%

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

   7. Applied egg-rr 12.7%

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

   8. Taylor expanded in kx around -inf 44.0%

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

      1. mul-1-neg 44.0%

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

      2. associate-*l/ 47.9%

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

      3. *-commutative 47.9%

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

      4. distribute-rgt-neg-in 47.9%

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

      5. *-lft-identity 47.9%

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

      6. metadata-eval 47.9%

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

      7. times-frac 47.9%

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

      8. neg-mul-1 47.9%

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

      9. neg-mul-1 47.9%

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

      10. distribute-frac-neg 47.9%

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

      11. remove-double-neg 47.9%

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

   10. Simplified 47.9%

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

   if -1.00000000000000004e-153 < (sin.f64 kx) < 1.00000000000000001e-150

   1. Initial program 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

      4. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 48.3%

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

   if 1.00000000000000001e-150 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 53.8%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-150}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 5: 45.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-150}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.02)
   (* (sin th) (fabs (/ (sin ky) (sin kx))))
   (if (<= (sin kx) -1e-153)
     (* (sin ky) (/ (sin th) (- kx)))
     (if (<= (sin kx) 1e-150) (sin th) (/ (sin th) (/ (sin kx) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.02) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else if (sin(kx) <= -1e-153) {
		tmp = sin(ky) * (sin(th) / -kx);
	} else if (sin(kx) <= 1e-150) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / sin(ky));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.02d0)) then
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    else if (sin(kx) <= (-1d-153)) then
        tmp = sin(ky) * (sin(th) / -kx)
    else if (sin(kx) <= 1d-150) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / sin(ky))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.02) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else if (Math.sin(kx) <= -1e-153) {
		tmp = Math.sin(ky) * (Math.sin(th) / -kx);
	} else if (Math.sin(kx) <= 1e-150) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.02:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	elif math.sin(kx) <= -1e-153:
		tmp = math.sin(ky) * (math.sin(th) / -kx)
	elif math.sin(kx) <= 1e-150:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.02)
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	elseif (sin(kx) <= -1e-153)
		tmp = Float64(sin(ky) * Float64(sin(th) / Float64(-kx)));
	elseif (sin(kx) <= 1e-150)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.02)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	elseif (sin(kx) <= -1e-153)
		tmp = sin(ky) * (sin(th) / -kx);
	elseif (sin(kx) <= 1e-150)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -1e-153], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / (-kx)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-150], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 15.9%

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

      1. add-sqr-sqrt 4.1%

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

      2. sqrt-unprod 36.2%

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

      3. pow2 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. unpow2 36.2%

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

      2. rem-sqrt-square 46.3%

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

   8. Simplified 46.3%

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

   if -0.0200000000000000004 < (sin.f64 kx) < -1.00000000000000004e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 7.0%

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

   5. Taylor expanded in kx around 0 7.1%

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

      1. add-sqr-sqrt 5.7%

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

      2. sqrt-unprod 14.9%

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

      3. pow2 14.9%

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

      4. associate-/l* 14.9%

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

   7. Applied egg-rr 14.9%

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

   8. Taylor expanded in kx around -inf 44.8%

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

      1. mul-1-neg 44.8%

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

      2. associate-*l/ 48.6%

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

      3. *-commutative 48.6%

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

      4. distribute-rgt-neg-in 48.6%

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

      5. *-lft-identity 48.6%

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

      6. metadata-eval 48.6%

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

      7. times-frac 48.6%

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

      8. neg-mul-1 48.6%

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

      9. neg-mul-1 48.6%

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

      10. distribute-frac-neg 48.6%

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

      11. remove-double-neg 48.6%

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

   10. Simplified 48.6%

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

   if -1.00000000000000004e-153 < (sin.f64 kx) < 1.00000000000000001e-150

   1. Initial program 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

      4. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 48.3%

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

   if 1.00000000000000001e-150 < (sin.f64 kx)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 53.8%

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

      1. *-commutative 53.8%

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

      2. clear-num 53.8%

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

      3. un-div-inv 53.8%

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

   6. Applied egg-rr 53.8%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 10^{-150}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 6: 60.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (sin th)
   (if (<= (sin th) 0.05)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (fabs (* (sin ky) (/ (sin th) (sin kx)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.05], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. unpow2 97.5%

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

      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 28.1%

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

   if -0.050000000000000003 < (sin.f64 th) < 0.050000000000000003

   1. Initial program 95.2%

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

      1. associate-*l/ 91.1%

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

      2. +-commutative 91.1%

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

      3. unpow2 91.1%

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

      4. unpow2 91.1%

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

      5. hypot-def 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in th around 0 90.6%

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

      1. expm1-log1p-u 90.5%

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

      2. expm1-udef 21.3%

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

      3. *-un-lft-identity 21.3%

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

      4. times-frac 21.3%

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

      5. /-rgt-identity 21.3%

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

   6. Applied egg-rr 21.3%

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

      1. expm1-def 98.0%

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

      2. expm1-log1p 98.1%

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

   8. Simplified 98.1%

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

   if 0.050000000000000003 < (sin.f64 th)

   1. Initial program 96.4%

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

      1. +-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. unpow2 96.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 22.7%

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

      1. add-sqr-sqrt 13.4%

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

      2. sqrt-unprod 17.2%

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

      3. pow2 17.2%

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

      4. associate-*l/ 17.2%

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

      5. *-un-lft-identity 17.2%

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

      6. times-frac 17.2%

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

      7. /-rgt-identity 17.2%

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

   6. Applied egg-rr 17.2%

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

      1. unpow2 17.2%

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

      2. rem-sqrt-square 28.5%

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

   8. Simplified 28.5%

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

4. Final simplification 66.0%

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

Alternative 7: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.05)
   (sin th)
   (if (<= (sin th) 0.05)
     (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
     (fabs (* (sin ky) (/ (sin th) (sin kx)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.05], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. unpow2 97.5%

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

      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 28.1%

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

   if -0.050000000000000003 < (sin.f64 th) < 0.050000000000000003

   1. Initial program 95.2%

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

      1. associate-*l/ 91.1%

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

      2. +-commutative 91.1%

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

      3. unpow2 91.1%

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

      4. unpow2 91.1%

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

      5. hypot-def 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in th around 0 90.6%

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

      1. associate-/l* 98.1%

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

      2. associate-/r/ 98.1%

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

   6. Applied egg-rr 98.1%

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

   if 0.050000000000000003 < (sin.f64 th)

   1. Initial program 96.4%

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

      1. +-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. unpow2 96.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 22.7%

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

      1. add-sqr-sqrt 13.4%

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

      2. sqrt-unprod 17.2%

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

      3. pow2 17.2%

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

      4. associate-*l/ 17.2%

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

      5. *-un-lft-identity 17.2%

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

      6. times-frac 17.2%

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

      7. /-rgt-identity 17.2%

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

   6. Applied egg-rr 17.2%

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

      1. unpow2 17.2%

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

      2. rem-sqrt-square 28.5%

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

   8. Simplified 28.5%

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

4. Final simplification 66.0%

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

Alternative 8: 42.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.0001)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) -1e-153)
     (* (sin ky) (/ (sin th) (- kx)))
     (if (<= (sin kx) 2e-44) (sin th) (/ (sin th) (/ (sin kx) ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.0001) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= -1e-153) {
		tmp = sin(ky) * (sin(th) / -kx);
	} else if (sin(kx) <= 2e-44) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (sin(kx) / ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.0001d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= (-1d-153)) then
        tmp = sin(ky) * (sin(th) / -kx)
    else if (sin(kx) <= 2d-44) then
        tmp = sin(th)
    else
        tmp = sin(th) / (sin(kx) / ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.0001) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= -1e-153) {
		tmp = Math.sin(ky) * (Math.sin(th) / -kx);
	} else if (Math.sin(kx) <= 2e-44) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.0001:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= -1e-153:
		tmp = math.sin(ky) * (math.sin(th) / -kx)
	elif math.sin(kx) <= 2e-44:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.0001)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= -1e-153)
		tmp = Float64(sin(ky) * Float64(sin(th) / Float64(-kx)));
	elseif (sin(kx) <= 2e-44)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.0001)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= -1e-153)
		tmp = sin(ky) * (sin(th) / -kx);
	elseif (sin(kx) <= 2e-44)
		tmp = sin(th);
	else
		tmp = sin(th) / (sin(kx) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.0001], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -1e-153], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / (-kx)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-44], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 16.0%

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

      1. associate-/l* 16.0%

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

   6. Simplified 16.0%

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

      1. add-sqr-sqrt 13.1%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

      4. div-inv 25.1%

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

      5. clear-num 25.1%

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

   8. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 35.4%

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

      3. associate-*r/ 35.4%

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

      4. associate-*l/ 35.4%

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

      5. *-commutative 35.4%

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

   10. Simplified 35.4%

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

   if -1.00000000000000005e-4 < (sin.f64 kx) < -1.00000000000000004e-153

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 7.2%

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

   5. Taylor expanded in kx around 0 7.3%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 12.7%

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

      3. pow2 12.7%

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

      4. associate-/l* 12.7%

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

   7. Applied egg-rr 12.7%

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

   8. Taylor expanded in kx around -inf 44.0%

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

      1. mul-1-neg 44.0%

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

      2. associate-*l/ 47.9%

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

      3. *-commutative 47.9%

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

      4. distribute-rgt-neg-in 47.9%

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

      5. *-lft-identity 47.9%

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

      6. metadata-eval 47.9%

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

      7. times-frac 47.9%

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

      8. neg-mul-1 47.9%

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

      9. neg-mul-1 47.9%

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

      10. distribute-frac-neg 47.9%

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

      11. remove-double-neg 47.9%

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

   10. Simplified 47.9%

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

   if -1.00000000000000004e-153 < (sin.f64 kx) < 1.99999999999999991e-44

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

      2. unpow2 89.7%

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

      3. unpow2 89.7%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 44.0%

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

   if 1.99999999999999991e-44 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 52.2%

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

      1. associate-/l* 52.2%

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

   6. Simplified 52.2%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0001:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{-kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]

Alternative 9: 47.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-297)
   (/ (sin ky) (/ (sin kx) (sin th)))
   (if (<= (sin ky) 4e-25) (/ (sin th) (fabs (/ (sin kx) ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-297) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (sin(ky) <= 4e-25) {
		tmp = sin(th) / fabs((sin(kx) / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-297)) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else if (sin(ky) <= 4d-25) then
        tmp = sin(th) / abs((sin(kx) / ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-297) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (Math.sin(ky) <= 4e-25) {
		tmp = Math.sin(th) / Math.abs((Math.sin(kx) / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-297:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif math.sin(ky) <= 4e-25:
		tmp = math.sin(th) / math.fabs((math.sin(kx) / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-297)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (sin(ky) <= 4e-25)
		tmp = Float64(sin(th) / abs(Float64(sin(kx) / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-297)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (sin(ky) <= 4e-25)
		tmp = sin(th) / abs((sin(kx) / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-297], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-25], N[(N[Sin[th], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.8%

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

      1. associate-/r/ 95.6%

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

      2. +-commutative 95.6%

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

      3. unpow2 95.6%

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

      4. unpow2 95.6%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 27.3%

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

   if -1.00000000000000004e-297 < (sin.f64 ky) < 4.00000000000000015e-25

   1. Initial program 92.6%

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

      1. +-commutative 92.6%

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

      2. unpow2 92.6%

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

      3. unpow2 92.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 49.3%

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

      1. associate-/l* 49.1%

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

   6. Simplified 49.1%

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

      1. add-sqr-sqrt 36.4%

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

      2. sqrt-unprod 61.4%

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

      3. pow2 61.4%

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

   8. Applied egg-rr 61.4%

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

      1. unpow2 61.4%

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

      2. rem-sqrt-square 78.9%

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

   10. Simplified 78.9%

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

   if 4.00000000000000015e-25 < (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 56.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 47.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-297)
   (/ (sin th) (/ (sin kx) (sin ky)))
   (if (<= (sin ky) 4e-25) (/ (sin th) (fabs (/ (sin kx) ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-297) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else if (sin(ky) <= 4e-25) {
		tmp = sin(th) / fabs((sin(kx) / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-297)) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else if (sin(ky) <= 4d-25) then
        tmp = sin(th) / abs((sin(kx) / ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-297) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else if (Math.sin(ky) <= 4e-25) {
		tmp = Math.sin(th) / Math.abs((Math.sin(kx) / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-297:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	elif math.sin(ky) <= 4e-25:
		tmp = math.sin(th) / math.fabs((math.sin(kx) / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-297)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	elseif (sin(ky) <= 4e-25)
		tmp = Float64(sin(th) / abs(Float64(sin(kx) / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-297)
		tmp = sin(th) / (sin(kx) / sin(ky));
	elseif (sin(ky) <= 4e-25)
		tmp = sin(th) / abs((sin(kx) / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-297], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-25], N[(N[Sin[th], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.8%

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

      1. +-commutative 95.8%

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

      2. unpow2 95.8%

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

      3. unpow2 95.8%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 27.2%

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

      1. *-commutative 27.2%

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

      2. clear-num 27.3%

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

      3. un-div-inv 27.3%

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

   6. Applied egg-rr 27.3%

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

   if -1.00000000000000004e-297 < (sin.f64 ky) < 4.00000000000000015e-25

   1. Initial program 92.6%

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

      1. +-commutative 92.6%

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

      2. unpow2 92.6%

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

      3. unpow2 92.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 49.3%

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

      1. associate-/l* 49.1%

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

   6. Simplified 49.1%

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

      1. add-sqr-sqrt 36.4%

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

      2. sqrt-unprod 61.4%

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

      3. pow2 61.4%

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

   8. Applied egg-rr 61.4%

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

      1. unpow2 61.4%

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

      2. rem-sqrt-square 78.9%

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

   10. Simplified 78.9%

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

   if 4.00000000000000015e-25 < (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 56.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 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 96.0%

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

   1. associate-*l/ 93.8%

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

   2. associate-*r/ 95.9%

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

   3. +-commutative 95.9%

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

   4. unpow2 95.9%

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

   5. unpow2 95.9%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 12: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -3.6 \cdot 10^{-6} \lor \neg \left(th \leq 1250000000000\right):\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (or (<= th -3.6e-6) (not (<= th 1250000000000.0)))
     (/ (* ky (sin th)) t_1)
     (* (/ (sin ky) t_1) th))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if ((th <= -3.6e-6) || !(th <= 1250000000000.0)) {
		tmp = (ky * sin(th)) / t_1;
	} else {
		tmp = (sin(ky) / t_1) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if ((th <= -3.6e-6) || !(th <= 1250000000000.0)) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else {
		tmp = (Math.sin(ky) / t_1) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if (th <= -3.6e-6) or not (th <= 1250000000000.0):
		tmp = (ky * math.sin(th)) / t_1
	else:
		tmp = (math.sin(ky) / t_1) * th
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if ((th <= -3.6e-6) || !(th <= 1250000000000.0))
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	else
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if ((th <= -3.6e-6) || ~((th <= 1250000000000.0)))
		tmp = (ky * sin(th)) / t_1;
	else
		tmp = (sin(ky) / t_1) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[th, -3.6e-6], N[Not[LessEqual[th, 1250000000000.0]], $MachinePrecision]], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < -3.59999999999999984e-6 or 1.25e12 < th

   1. Initial program 96.9%

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

      1. associate-*l/ 96.9%

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

      2. +-commutative 96.9%

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

      3. unpow2 96.9%

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

      4. unpow2 96.9%

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

      5. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 50.6%

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

   if -3.59999999999999984e-6 < th < 1.25e12

   1. Initial program 95.2%

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

      1. associate-*l/ 91.1%

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

      2. +-commutative 91.1%

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

      3. unpow2 91.1%

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

      4. unpow2 91.1%

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

      5. hypot-def 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in th around 0 90.2%

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

      1. associate-/l* 97.7%

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

      2. associate-/r/ 97.8%

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

   6. Applied egg-rr 97.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -3.6 \cdot 10^{-6} \lor \neg \left(th \leq 1250000000000\right):\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \end{array} \]

Alternative 13: 40.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-120}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-120) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-120) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-120) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-120) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-120:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-120)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-120)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-120], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.99999999999999979e-121

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. unpow2 94.2%

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

      3. unpow2 94.2%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 33.3%

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

   if 9.99999999999999979e-121 < (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 53.0%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-120}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 40.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-120}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-120) (/ (sin th) (/ (sin kx) ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-120) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-120) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-120) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-120:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-120)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-120)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-120], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.99999999999999979e-121

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. unpow2 94.2%

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

      3. unpow2 94.2%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 32.2%

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

      1. associate-/l* 33.3%

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

   6. Simplified 33.3%

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

   if 9.99999999999999979e-121 < (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 53.0%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-120}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 34.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-128) (/ (sin th) (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-128) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-128) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-128) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-128:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-128)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-128)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-128], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2.00000000000000011e-128

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. unpow2 94.1%

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

      3. unpow2 94.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 35.1%

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

   5. Taylor expanded in kx around 0 23.5%

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

   6. Taylor expanded in ky around 0 23.3%

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

      1. associate-/l* 24.5%

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

   8. Simplified 24.5%

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

   if 2.00000000000000011e-128 < (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 53.0%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 32.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -470000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.4 \cdot 10^{-173}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -470000.0)
   (sin th)
   (if (<= ky 1.4e-173) (* ky (/ th (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -470000.0) {
		tmp = sin(th);
	} else if (ky <= 1.4e-173) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-470000.0d0)) then
        tmp = sin(th)
    else if (ky <= 1.4d-173) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -470000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1.4e-173) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -470000.0:
		tmp = math.sin(th)
	elif ky <= 1.4e-173:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -470000.0)
		tmp = sin(th);
	elseif (ky <= 1.4e-173)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -470000.0)
		tmp = sin(th);
	elseif (ky <= 1.4e-173)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -470000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.4e-173], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -4.7e5 or 1.39999999999999995e-173 < ky

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. hypot-def 99.7%

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

   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 30.4%

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

   if -4.7e5 < ky < 1.39999999999999995e-173

   1. Initial program 89.6%

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

      1. associate-*l/ 85.5%

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

      2. +-commutative 85.5%

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

      3. unpow2 85.5%

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

      4. unpow2 85.5%

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

      5. hypot-def 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in th around 0 47.2%

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

   5. Taylor expanded in ky around 0 34.0%

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

      1. associate-/l* 36.3%

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

   7. Simplified 36.3%

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

      1. clear-num 36.2%

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

      2. associate-/r/ 36.2%

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

      3. clear-num 36.2%

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

   9. Applied egg-rr 36.2%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -470000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.4 \cdot 10^{-173}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 32.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -470000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -470000.0)
   (sin th)
   (if (<= ky 1.8e-172) (/ ky (/ (sin kx) th)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -470000.0) {
		tmp = sin(th);
	} else if (ky <= 1.8e-172) {
		tmp = ky / (sin(kx) / th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-470000.0d0)) then
        tmp = sin(th)
    else if (ky <= 1.8d-172) then
        tmp = ky / (sin(kx) / th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -470000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1.8e-172) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -470000.0:
		tmp = math.sin(th)
	elif ky <= 1.8e-172:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -470000.0)
		tmp = sin(th);
	elseif (ky <= 1.8e-172)
		tmp = Float64(ky / Float64(sin(kx) / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -470000.0)
		tmp = sin(th);
	elseif (ky <= 1.8e-172)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -470000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.8e-172], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -4.7e5 or 1.80000000000000007e-172 < ky

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. hypot-def 99.7%

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

   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 30.4%

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

   if -4.7e5 < ky < 1.80000000000000007e-172

   1. Initial program 89.6%

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

      1. associate-*l/ 85.5%

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

      2. +-commutative 85.5%

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

      3. unpow2 85.5%

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

      4. unpow2 85.5%

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

      5. hypot-def 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in th around 0 47.2%

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

   5. Taylor expanded in ky around 0 34.0%

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

      1. associate-/l* 36.3%

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

   7. Simplified 36.3%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -470000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18: 31.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -2900000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.45 \cdot 10^{-163}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -2900000.0)
   (sin th)
   (if (<= ky 2.45e-163) (* th (/ ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -2900000.0) {
		tmp = sin(th);
	} else if (ky <= 2.45e-163) {
		tmp = th * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-2900000.0d0)) then
        tmp = sin(th)
    else if (ky <= 2.45d-163) then
        tmp = th * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -2900000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 2.45e-163) {
		tmp = th * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -2900000.0:
		tmp = math.sin(th)
	elif ky <= 2.45e-163:
		tmp = th * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -2900000.0)
		tmp = sin(th);
	elseif (ky <= 2.45e-163)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -2900000.0)
		tmp = sin(th);
	elseif (ky <= 2.45e-163)
		tmp = th * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -2900000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 2.45e-163], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -2.9e6 or 2.4500000000000001e-163 < ky

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. hypot-def 99.7%

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

   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 30.8%

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

   if -2.9e6 < ky < 2.4500000000000001e-163

   1. Initial program 89.9%

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

      1. associate-*l/ 84.9%

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

      2. +-commutative 84.9%

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

      3. unpow2 84.9%

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

      4. unpow2 84.9%

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

      5. hypot-def 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in th around 0 45.8%

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

   5. Taylor expanded in ky around 0 33.0%

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

      1. associate-/l* 35.2%

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

   7. Simplified 35.2%

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

   8. Taylor expanded in kx around 0 32.6%

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

      1. associate-/r/ 32.9%

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

   10. Applied egg-rr 32.9%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2900000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.45 \cdot 10^{-163}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19: 21.9% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -125000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2 \cdot 10^{-120}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -125000.0) th (if (<= ky 2e-120) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -125000.0) {
		tmp = th;
	} else if (ky <= 2e-120) {
		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 <= (-125000.0d0)) then
        tmp = th
    else if (ky <= 2d-120) 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 <= -125000.0) {
		tmp = th;
	} else if (ky <= 2e-120) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -125000.0:
		tmp = th
	elif ky <= 2e-120:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -125000.0)
		tmp = th;
	elseif (ky <= 2e-120)
		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 <= -125000.0)
		tmp = th;
	elseif (ky <= 2e-120)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -125000.0], th, If[LessEqual[ky, 2e-120], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -125000 or 1.99999999999999996e-120 < 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/ 98.9%

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

      2. +-commutative 98.9%

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

      3. unpow2 98.9%

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

      4. unpow2 98.9%

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

      5. hypot-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in th around 0 53.1%

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

   5. Taylor expanded in kx around 0 17.7%

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

   if -125000 < ky < 1.99999999999999996e-120

   1. Initial program 90.5%

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

      1. associate-*l/ 86.1%

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

      2. +-commutative 86.1%

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

      3. unpow2 86.1%

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

      4. unpow2 86.1%

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

      5. hypot-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in th around 0 46.5%

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

   5. Taylor expanded in ky around 0 33.3%

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

      1. associate-/l* 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in kx around 0 32.1%

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

      1. associate-/r/ 32.4%

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

   10. Applied egg-rr 32.4%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -125000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2 \cdot 10^{-120}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 20: 13.5% 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 96.0%

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

   1. associate-*l/ 93.8%

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

   2. +-commutative 93.8%

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

   3. unpow2 93.8%

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

   4. unpow2 93.8%

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

   5. hypot-def 95.6%

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

3. Simplified 95.6%

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

4. Taylor expanded in th around 0 50.4%

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

5. Taylor expanded in kx around 0 13.2%

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

6. Final simplification 13.2%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023182 
(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)))