Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 24.6s

Alternatives: 24

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% 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 93.9%

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

   1. +-commutative 93.9%

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

   2. unpow2 93.9%

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

   3. unpow2 93.9%

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

   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: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-10}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-24}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ ky (hypot ky kx)))))
   (if (<= (sin ky) -1e-10)
     (- (sin th))
     (if (<= (sin ky) -5e-303)
       t_1
       (if (<= (sin ky) 5e-93)
         (/ (sin th) (/ (sin kx) ky))
         (if (<= (sin ky) 1e-49)
           t_1
           (if (<= (sin ky) 2e-24) (* ky (/ (sin th) (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (ky / hypot(ky, kx));
	double tmp;
	if (sin(ky) <= -1e-10) {
		tmp = -sin(th);
	} else if (sin(ky) <= -5e-303) {
		tmp = t_1;
	} else if (sin(ky) <= 5e-93) {
		tmp = sin(th) / (sin(kx) / ky);
	} else if (sin(ky) <= 1e-49) {
		tmp = t_1;
	} else if (sin(ky) <= 2e-24) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) * (ky / Math.hypot(ky, kx));
	double tmp;
	if (Math.sin(ky) <= -1e-10) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= -5e-303) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 5e-93) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else if (Math.sin(ky) <= 1e-49) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 2e-24) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) * (ky / math.hypot(ky, kx))
	tmp = 0
	if math.sin(ky) <= -1e-10:
		tmp = -math.sin(th)
	elif math.sin(ky) <= -5e-303:
		tmp = t_1
	elif math.sin(ky) <= 5e-93:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	elif math.sin(ky) <= 1e-49:
		tmp = t_1
	elif math.sin(ky) <= 2e-24:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(ky / hypot(ky, kx)))
	tmp = 0.0
	if (sin(ky) <= -1e-10)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= -5e-303)
		tmp = t_1;
	elseif (sin(ky) <= 5e-93)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	elseif (sin(ky) <= 1e-49)
		tmp = t_1;
	elseif (sin(ky) <= 2e-24)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) * (ky / hypot(ky, kx));
	tmp = 0.0;
	if (sin(ky) <= -1e-10)
		tmp = -sin(th);
	elseif (sin(ky) <= -5e-303)
		tmp = t_1;
	elseif (sin(ky) <= 5e-93)
		tmp = sin(th) / (sin(kx) / ky);
	elseif (sin(ky) <= 1e-49)
		tmp = t_1;
	elseif (sin(ky) <= 2e-24)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-10], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-303], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-93], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-49], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-24], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -1.00000000000000004e-10

   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 11.0%

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

   5. Taylor expanded in ky around 0 32.3%

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

   6. Taylor expanded in ky around -inf 54.1%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 54.1%

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

   8. Simplified 54.1%

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

   if -1.00000000000000004e-10 < (sin.f64 ky) < -4.9999999999999998e-303 or 4.99999999999999994e-93 < (sin.f64 ky) < 9.99999999999999936e-50

   1. Initial program 87.7%

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

      1. +-commutative 87.7%

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

      2. unpow2 87.7%

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

      3. unpow2 87.7%

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

      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 ky around 0 99.7%

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

   5. Taylor expanded in ky around 0 99.7%

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

   6. Taylor expanded in kx around 0 66.7%

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

   if -4.9999999999999998e-303 < (sin.f64 ky) < 4.99999999999999994e-93

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. unpow2 86.5%

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

      3. unpow2 86.5%

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

      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 ky around 0 53.2%

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

      1. associate-/l* 56.0%

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

   6. Simplified 56.0%

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

   if 9.99999999999999936e-50 < (sin.f64 ky) < 1.99999999999999985e-24

   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 35.3%

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

      1. associate-/l* 35.3%

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

      2. associate-/r/ 35.8%

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

   6. Applied egg-rr 35.8%

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

   if 1.99999999999999985e-24 < (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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 58.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-10}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-49}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-24}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 53.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ t_2 := \sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))) (t_2 (* (sin th) (/ ky (sin kx)))))
   (if (<= (sin ky) -1.6e-33)
     t_1
     (if (<= (sin ky) -5e-196)
       t_2
       (if (<= (sin ky) -1e-230) t_1 (if (<= (sin ky) 2e-24) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double t_2 = sin(th) * (ky / sin(kx));
	double tmp;
	if (sin(ky) <= -1.6e-33) {
		tmp = t_1;
	} else if (sin(ky) <= -5e-196) {
		tmp = t_2;
	} else if (sin(ky) <= -1e-230) {
		tmp = t_1;
	} else if (sin(ky) <= 2e-24) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -sin(th)
    t_2 = sin(th) * (ky / sin(kx))
    if (sin(ky) <= (-1.6d-33)) then
        tmp = t_1
    else if (sin(ky) <= (-5d-196)) then
        tmp = t_2
    else if (sin(ky) <= (-1d-230)) then
        tmp = t_1
    else if (sin(ky) <= 2d-24) then
        tmp = t_2
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double t_2 = Math.sin(th) * (ky / Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -1.6e-33) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -5e-196) {
		tmp = t_2;
	} else if (Math.sin(ky) <= -1e-230) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 2e-24) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	t_2 = math.sin(th) * (ky / math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -1.6e-33:
		tmp = t_1
	elif math.sin(ky) <= -5e-196:
		tmp = t_2
	elif math.sin(ky) <= -1e-230:
		tmp = t_1
	elif math.sin(ky) <= 2e-24:
		tmp = t_2
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	t_2 = Float64(sin(th) * Float64(ky / sin(kx)))
	tmp = 0.0
	if (sin(ky) <= -1.6e-33)
		tmp = t_1;
	elseif (sin(ky) <= -5e-196)
		tmp = t_2;
	elseif (sin(ky) <= -1e-230)
		tmp = t_1;
	elseif (sin(ky) <= 2e-24)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	t_2 = sin(th) * (ky / sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -1.6e-33)
		tmp = t_1;
	elseif (sin(ky) <= -5e-196)
		tmp = t_2;
	elseif (sin(ky) <= -1e-230)
		tmp = t_1;
	elseif (sin(ky) <= 2e-24)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, Block[{t$95$2 = N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1.6e-33], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-196], t$95$2, If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-230], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-24], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1.59999999999999988e-33 or -5.0000000000000005e-196 < (sin.f64 ky) < -1.00000000000000005e-230

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. unpow2 93.6%

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

      3. unpow2 93.6%

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

      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 25.3%

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

   5. Taylor expanded in ky around 0 43.2%

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

   6. Taylor expanded in ky around -inf 58.1%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 58.1%

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

   8. Simplified 58.1%

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

   if -1.59999999999999988e-33 < (sin.f64 ky) < -5.0000000000000005e-196 or -1.00000000000000005e-230 < (sin.f64 ky) < 1.99999999999999985e-24

   1. Initial program 90.5%

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

      1. +-commutative 90.5%

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

      2. unpow2 90.5%

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

      3. unpow2 90.5%

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

      4. hypot-def 99.6%

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

   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 52.8%

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

   if 1.99999999999999985e-24 < (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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 58.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-230}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 53.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;\sin ky \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-196}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= (sin ky) -1.6e-33)
     t_1
     (if (<= (sin ky) -5e-196)
       (* ky (/ (sin th) (sin kx)))
       (if (<= (sin ky) -1e-230)
         t_1
         (if (<= (sin ky) 2e-24) (* (sin th) (/ ky (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (sin(ky) <= -1.6e-33) {
		tmp = t_1;
	} else if (sin(ky) <= -5e-196) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (sin(ky) <= -1e-230) {
		tmp = t_1;
	} else if (sin(ky) <= 2e-24) {
		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) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (sin(ky) <= (-1.6d-33)) then
        tmp = t_1
    else if (sin(ky) <= (-5d-196)) then
        tmp = ky * (sin(th) / sin(kx))
    else if (sin(ky) <= (-1d-230)) then
        tmp = t_1
    else if (sin(ky) <= 2d-24) 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 t_1 = -Math.sin(th);
	double tmp;
	if (Math.sin(ky) <= -1.6e-33) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -5e-196) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(ky) <= -1e-230) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 2e-24) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if math.sin(ky) <= -1.6e-33:
		tmp = t_1
	elif math.sin(ky) <= -5e-196:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif math.sin(ky) <= -1e-230:
		tmp = t_1
	elif math.sin(ky) <= 2e-24:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (sin(ky) <= -1.6e-33)
		tmp = t_1;
	elseif (sin(ky) <= -5e-196)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= -1e-230)
		tmp = t_1;
	elseif (sin(ky) <= 2e-24)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (sin(ky) <= -1.6e-33)
		tmp = t_1;
	elseif (sin(ky) <= -5e-196)
		tmp = ky * (sin(th) / sin(kx));
	elseif (sin(ky) <= -1e-230)
		tmp = t_1;
	elseif (sin(ky) <= 2e-24)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1.6e-33], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-196], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-230], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-24], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1.59999999999999988e-33 or -5.0000000000000005e-196 < (sin.f64 ky) < -1.00000000000000005e-230

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. unpow2 93.6%

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

      3. unpow2 93.6%

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

      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 25.3%

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

   5. Taylor expanded in ky around 0 43.2%

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

   6. Taylor expanded in ky around -inf 58.1%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 58.1%

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

   8. Simplified 58.1%

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

   if -1.59999999999999988e-33 < (sin.f64 ky) < -5.0000000000000005e-196

   1. Initial program 95.9%

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

      1. +-commutative 95.9%

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

      2. unpow2 95.9%

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

      3. unpow2 95.9%

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

      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 53.5%

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

      1. associate-/l* 53.5%

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

      2. associate-/r/ 53.5%

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

   6. Applied egg-rr 53.5%

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

   if -1.00000000000000005e-230 < (sin.f64 ky) < 1.99999999999999985e-24

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. unpow2 88.6%

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

      3. unpow2 88.6%

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

      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 52.6%

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

   if 1.99999999999999985e-24 < (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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 58.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1.6 \cdot 10^{-33}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-196}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-230}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-5}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{ky \cdot ky}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-5)
   (- (sin th))
   (if (<= (sin ky) 0.0005)
     (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
     (* (sin th) (+ 1.0 (* -0.5 (/ (pow (sin kx) 2.0) (* ky ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-5) {
		tmp = -sin(th);
	} else if (sin(ky) <= 0.0005) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th) * (1.0 + (-0.5 * (pow(sin(kx), 2.0) / (ky * ky))));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-5) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 0.0005) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * (1.0 + (-0.5 * (Math.pow(Math.sin(kx), 2.0) / (ky * ky))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-5:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 0.0005:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th) * (1.0 + (-0.5 * (math.pow(math.sin(kx), 2.0) / (ky * ky))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-5)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 0.0005)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(1.0 + Float64(-0.5 * Float64((sin(kx) ^ 2.0) / Float64(ky * ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-5)
		tmp = -sin(th);
	elseif (sin(ky) <= 0.0005)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th) * (1.0 + (-0.5 * ((sin(kx) ^ 2.0) / (ky * ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-5], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0005], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1.00000000000000008e-5

   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 99.6%

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

   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 6.9%

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

   5. Taylor expanded in ky around 0 29.2%

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

   6. Taylor expanded in ky around -inf 54.7%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 54.7%

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

   8. Simplified 54.7%

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

   if -1.00000000000000008e-5 < (sin.f64 ky) < 5.0000000000000001e-4

   1. Initial program 88.4%

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

      1. associate-*l/ 85.3%

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

      2. associate-*r/ 88.4%

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

      3. +-commutative 88.4%

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

      4. unpow2 88.4%

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

      5. unpow2 88.4%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 99.5%

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

   if 5.0000000000000001e-4 < (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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in ky around 0 6.0%

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

   5. Taylor expanded in ky around 0 33.0%

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

   6. Taylor expanded in ky around inf 58.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{{ky}^{2}}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{\color{blue}{ky \cdot ky}}\right) \cdot \sin th \]

   8. Simplified 58.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{ky \cdot ky}\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-5}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{ky \cdot ky}\right)\\ \end{array} \]

Alternative 6: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{ky \cdot ky}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (- (sin th))
   (if (<= (sin ky) 0.0005)
     (* (sin th) (/ ky (hypot ky (sin kx))))
     (* (sin th) (+ 1.0 (* -0.5 (/ (pow (sin kx) 2.0) (* ky ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = -sin(th);
	} else if (sin(ky) <= 0.0005) {
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th) * (1.0 + (-0.5 * (pow(sin(kx), 2.0) / (ky * ky))));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 0.0005) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * (1.0 + (-0.5 * (Math.pow(Math.sin(kx), 2.0) / (ky * ky))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 0.0005:
		tmp = math.sin(th) * (ky / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th) * (1.0 + (-0.5 * (math.pow(math.sin(kx), 2.0) / (ky * ky))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 0.0005)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(1.0 + Float64(-0.5 * Float64((sin(kx) ^ 2.0) / Float64(ky * ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = -sin(th);
	elseif (sin(ky) <= 0.0005)
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	else
		tmp = sin(th) * (1.0 + (-0.5 * ((sin(kx) ^ 2.0) / (ky * ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0005], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   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 99.6%

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

   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 5.9%

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

   5. Taylor expanded in ky around 0 28.1%

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

   6. Taylor expanded in ky around -inf 54.0%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 54.0%

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

   8. Simplified 54.0%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 5.0000000000000001e-4

   1. Initial program 88.5%

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

      1. +-commutative 88.5%

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

      2. unpow2 88.5%

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

      3. unpow2 88.5%

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

      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 ky around 0 99.4%

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

   5. Taylor expanded in ky around 0 99.4%

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

   if 5.0000000000000001e-4 < (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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in ky around 0 6.0%

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

   5. Taylor expanded in ky around 0 33.0%

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

   6. Taylor expanded in ky around inf 58.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{{ky}^{2}}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{\color{blue}{ky \cdot ky}}\right) \cdot \sin th \]

   8. Simplified 58.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{ky \cdot ky}\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 + -0.5 \cdot \frac{{\sin kx}^{2}}{ky \cdot ky}\right)\\ \end{array} \]

Alternative 7: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (- (sin th))
   (if (<= (sin ky) 0.0005) (* (sin th) (/ ky (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = -sin(th);
	} else if (sin(ky) <= 0.0005) {
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 0.0005) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 0.0005:
		tmp = math.sin(th) * (ky / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 0.0005)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = -sin(th);
	elseif (sin(ky) <= 0.0005)
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0005], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   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 99.6%

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

   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 5.9%

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

   5. Taylor expanded in ky around 0 28.1%

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

   6. Taylor expanded in ky around -inf 54.0%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 54.0%

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

   8. Simplified 54.0%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 5.0000000000000001e-4

   1. Initial program 88.5%

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

      1. +-commutative 88.5%

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

      2. unpow2 88.5%

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

      3. unpow2 88.5%

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

      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 ky around 0 99.4%

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

   5. Taylor expanded in ky around 0 99.4%

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

   if 5.0000000000000001e-4 < (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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 58.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (* (sin th) (/ ky (fabs (sin kx))))
   (if (<= (sin kx) 1e-6)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin ky) (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else if (sin(kx) <= 1e-6) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.04:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 1e-6:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.04)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	elseif (sin(kx) <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.04)
		tmp = sin(th) * (ky / abs(sin(kx)));
	elseif (sin(kx) <= 1e-6)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   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 99.6%

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

   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 20.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 62.9%

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

      3. rem-sqrt-square 62.9%

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

   6. Applied egg-rr 62.9%

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

   7. Taylor expanded in ky around 0 54.4%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 9.99999999999999955e-7

   1. Initial program 87.4%

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

      1. +-commutative 87.4%

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

      2. unpow2 87.4%

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

      3. unpow2 87.4%

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

      4. hypot-def 99.9%

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

   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 ky around 0 55.4%

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

   5. Taylor expanded in ky around 0 76.9%

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

   6. Taylor expanded in kx around 0 76.2%

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

   if 9.99999999999999955e-7 < (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. associate-*l/ 99.3%

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

      2. associate-*r/ 99.3%

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

      3. +-commutative 99.3%

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

      4. unpow2 99.3%

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

      5. unpow2 99.3%

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

      6. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in ky around 0 57.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 9: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (* (sin th) (/ ky (fabs (sin kx))))
   (if (<= (sin kx) 1e-6)
     (* (sin th) (/ ky (hypot ky kx)))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else if (sin(kx) <= 1e-6) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.04:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 1e-6:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.04)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	elseif (sin(kx) <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.04)
		tmp = sin(th) * (ky / abs(sin(kx)));
	elseif (sin(kx) <= 1e-6)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   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 99.6%

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

   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 20.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 62.9%

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

      3. rem-sqrt-square 62.9%

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

   6. Applied egg-rr 62.9%

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

   7. Taylor expanded in ky around 0 54.4%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 9.99999999999999955e-7

   1. Initial program 87.4%

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

      1. +-commutative 87.4%

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

      2. unpow2 87.4%

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

      3. unpow2 87.4%

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

      4. hypot-def 99.9%

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

   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 ky around 0 55.4%

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

   5. Taylor expanded in ky around 0 76.9%

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

   6. Taylor expanded in kx around 0 76.2%

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

   if 9.99999999999999955e-7 < (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 57.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 10: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (* (sin th) (/ ky (fabs (sin kx))))
   (if (<= (sin kx) 1e-6)
     (* (sin th) (/ ky (hypot ky kx)))
     (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else if (sin(kx) <= 1e-6) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else if (Math.sin(kx) <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} 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) <= -0.04:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	elif math.sin(kx) <= 1e-6:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	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) <= -0.04)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	elseif (sin(kx) <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	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) <= -0.04)
		tmp = sin(th) * (ky / abs(sin(kx)));
	elseif (sin(kx) <= 1e-6)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   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 99.6%

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

   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 20.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 62.9%

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

      3. rem-sqrt-square 62.9%

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

   6. Applied egg-rr 62.9%

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

   7. Taylor expanded in ky around 0 54.4%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 9.99999999999999955e-7

   1. Initial program 87.4%

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

      1. +-commutative 87.4%

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

      2. unpow2 87.4%

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

      3. unpow2 87.4%

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

      4. hypot-def 99.9%

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

   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 ky around 0 55.4%

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

   5. Taylor expanded in ky around 0 76.9%

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

   6. Taylor expanded in kx around 0 76.2%

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

   if 9.99999999999999955e-7 < (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. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 57.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 11: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{ky}}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.04)
   (/ (sin th) (/ (fabs (sin kx)) ky))
   (if (<= (sin kx) 1e-6)
     (* (sin th) (/ ky (hypot ky kx)))
     (/ (sin ky) (/ (sin kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.04) {
		tmp = sin(th) / (fabs(sin(kx)) / ky);
	} else if (sin(kx) <= 1e-6) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.04) {
		tmp = Math.sin(th) / (Math.abs(Math.sin(kx)) / ky);
	} else if (Math.sin(kx) <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} 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) <= -0.04:
		tmp = math.sin(th) / (math.fabs(math.sin(kx)) / ky)
	elif math.sin(kx) <= 1e-6:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	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) <= -0.04)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / ky));
	elseif (sin(kx) <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	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) <= -0.04)
		tmp = sin(th) / (abs(sin(kx)) / ky);
	elseif (sin(kx) <= 1e-6)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0400000000000000008

   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 99.6%

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

   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 20.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 62.9%

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

      3. rem-sqrt-square 62.9%

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

   6. Applied egg-rr 62.9%

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

   7. Taylor expanded in ky around 0 54.3%

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

      1. associate-/l* 54.3%

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

   9. Simplified 54.3%

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

   if -0.0400000000000000008 < (sin.f64 kx) < 9.99999999999999955e-7

   1. Initial program 87.4%

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

      1. +-commutative 87.4%

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

      2. unpow2 87.4%

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

      3. unpow2 87.4%

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

      4. hypot-def 99.9%

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

   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 ky around 0 55.4%

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

   5. Taylor expanded in ky around 0 76.9%

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

   6. Taylor expanded in kx around 0 76.2%

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

   if 9.99999999999999955e-7 < (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. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 57.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{ky}}\\ \mathbf{elif}\;\sin kx \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 12: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. associate-*l/ 92.3%

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

   2. associate-*r/ 93.8%

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

   3. +-commutative 93.8%

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

   4. unpow2 93.8%

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

   5. unpow2 93.8%

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

   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 13: 79.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(ky, \sin kx\right)\\ \mathbf{if}\;th \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;th \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot ky (sin kx))))
   (if (<= th -8.2e-10)
     (* (sin th) (/ ky t_1))
     (if (<= th 1.35e+15)
       (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
       (if (<= th 4.2e+151)
         (/ (sin th) (/ t_1 ky))
         (* (sin ky) (/ (sin th) (hypot (sin ky) kx))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(ky, sin(kx));
	double tmp;
	if (th <= -8.2e-10) {
		tmp = sin(th) * (ky / t_1);
	} else if (th <= 1.35e+15) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (th <= 4.2e+151) {
		tmp = sin(th) / (t_1 / ky);
	} else {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(ky, Math.sin(kx));
	double tmp;
	if (th <= -8.2e-10) {
		tmp = Math.sin(th) * (ky / t_1);
	} else if (th <= 1.35e+15) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (th <= 4.2e+151) {
		tmp = Math.sin(th) / (t_1 / ky);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(ky, math.sin(kx))
	tmp = 0
	if th <= -8.2e-10:
		tmp = math.sin(th) * (ky / t_1)
	elif th <= 1.35e+15:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif th <= 4.2e+151:
		tmp = math.sin(th) / (t_1 / ky)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(ky, sin(kx))
	tmp = 0.0
	if (th <= -8.2e-10)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	elseif (th <= 1.35e+15)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (th <= 4.2e+151)
		tmp = Float64(sin(th) / Float64(t_1 / ky));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(ky, sin(kx));
	tmp = 0.0;
	if (th <= -8.2e-10)
		tmp = sin(th) * (ky / t_1);
	elseif (th <= 1.35e+15)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (th <= 4.2e+151)
		tmp = sin(th) / (t_1 / ky);
	else
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, -8.2e-10], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.35e+15], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 4.2e+151], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / ky), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if th < -8.1999999999999996e-10

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. unpow2 92.4%

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

      3. unpow2 92.4%

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

      4. hypot-def 99.7%

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

   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 51.7%

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

   5. Taylor expanded in ky around 0 63.3%

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

   if -8.1999999999999996e-10 < th < 1.35e15

   1. Initial program 93.7%

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

      1. associate-/r/ 93.7%

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

      2. +-commutative 93.7%

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

      3. unpow2 93.7%

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

      4. unpow2 93.7%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 92.2%

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

      1. associate-*r/ 92.3%

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

      2. unpow2 92.3%

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

      3. unpow2 92.3%

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

      4. hypot-def 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. hypot-def 92.3%

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

      7. unpow2 92.3%

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

      8. unpow2 92.3%

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

      9. +-commutative 92.3%

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

      10. unpow2 92.3%

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

      11. unpow2 92.3%

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

      12. hypot-def 98.3%

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

   6. Simplified 98.3%

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

      1. clear-num 97.7%

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

      2. associate-/r/ 98.1%

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

      3. clear-num 98.3%

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

      4. hypot-udef 92.3%

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

      5. +-commutative 92.3%

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

      6. hypot-udef 98.3%

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

   8. Applied egg-rr 98.3%

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

   if 1.35e15 < th < 4.2000000000000001e151

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. unpow2 95.5%

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

      3. unpow2 95.5%

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

      4. hypot-def 99.4%

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

   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 63.5%

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

   5. Taylor expanded in ky around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. clear-num 75.9%

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

      3. un-div-inv 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.2000000000000001e151 < th

   1. Initial program 96.5%

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

      1. associate-*l/ 96.4%

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

      2. associate-*r/ 96.3%

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

      3. +-commutative 96.3%

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

      4. unpow2 96.3%

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

      5. unpow2 96.3%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 67.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]

Alternative 14: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(ky, \sin kx\right)\\ \mathbf{if}\;th \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;th \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;th \leq 1.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot ky (sin kx))))
   (if (<= th -7.8e-5)
     (* (sin th) (/ ky t_1))
     (if (<= th 1.35e+15)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= th 1.9e+151)
         (/ (sin th) (/ t_1 ky))
         (* (sin ky) (/ (sin th) (hypot (sin ky) kx))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(ky, sin(kx));
	double tmp;
	if (th <= -7.8e-5) {
		tmp = sin(th) * (ky / t_1);
	} else if (th <= 1.35e+15) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (th <= 1.9e+151) {
		tmp = sin(th) / (t_1 / ky);
	} else {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(ky, Math.sin(kx));
	double tmp;
	if (th <= -7.8e-5) {
		tmp = Math.sin(th) * (ky / t_1);
	} else if (th <= 1.35e+15) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else if (th <= 1.9e+151) {
		tmp = Math.sin(th) / (t_1 / ky);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(ky, math.sin(kx))
	tmp = 0
	if th <= -7.8e-5:
		tmp = math.sin(th) * (ky / t_1)
	elif th <= 1.35e+15:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	elif th <= 1.9e+151:
		tmp = math.sin(th) / (t_1 / ky)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(ky, sin(kx))
	tmp = 0.0
	if (th <= -7.8e-5)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	elseif (th <= 1.35e+15)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (th <= 1.9e+151)
		tmp = Float64(sin(th) / Float64(t_1 / ky));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(ky, sin(kx));
	tmp = 0.0;
	if (th <= -7.8e-5)
		tmp = sin(th) * (ky / t_1);
	elseif (th <= 1.35e+15)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	elseif (th <= 1.9e+151)
		tmp = sin(th) / (t_1 / ky);
	else
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, -7.8e-5], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.35e+15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[th, 1.9e+151], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / ky), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if th < -7.7999999999999999e-5

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. unpow2 92.2%

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

      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 50.2%

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

   5. Taylor expanded in ky around 0 62.1%

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

   if -7.7999999999999999e-5 < th < 1.35e15

   1. Initial program 93.8%

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

      1. associate-/r/ 93.8%

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

      2. +-commutative 93.8%

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

      3. unpow2 93.8%

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

      4. unpow2 93.8%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 92.3%

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

      1. associate-*r/ 92.4%

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

      2. unpow2 92.4%

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

      3. unpow2 92.4%

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

      4. hypot-def 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. hypot-def 92.4%

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

      7. unpow2 92.4%

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

      8. unpow2 92.4%

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

      9. +-commutative 92.4%

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

      10. unpow2 92.4%

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

      11. unpow2 92.4%

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

      12. hypot-def 98.3%

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

   6. Simplified 98.3%

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

      1. associate-/r/ 98.4%

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

      2. hypot-udef 92.4%

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

      3. +-commutative 92.4%

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

      4. hypot-udef 98.4%

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

   8. Applied egg-rr 98.4%

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

   if 1.35e15 < th < 1.9e151

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. unpow2 95.5%

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

      3. unpow2 95.5%

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

      4. hypot-def 99.4%

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

   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 63.5%

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

   5. Taylor expanded in ky around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. clear-num 75.9%

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

      3. un-div-inv 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 1.9e151 < th

   1. Initial program 96.5%

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

      1. associate-*l/ 96.4%

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

      2. associate-*r/ 96.3%

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

      3. +-commutative 96.3%

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

      4. unpow2 96.3%

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

      5. unpow2 96.3%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 67.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;th \leq 1.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]

Alternative 15: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(ky, \sin kx\right)\\ \mathbf{if}\;th \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;th \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;th \leq 4.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot ky (sin kx))))
   (if (<= th -2.3e-5)
     (* (sin th) (/ ky t_1))
     (if (<= th 1.35e+15)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= th 4.1e+151)
         (/ (sin th) (/ t_1 ky))
         (/ (sin th) (/ (hypot (sin ky) kx) (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(ky, sin(kx));
	double tmp;
	if (th <= -2.3e-5) {
		tmp = sin(th) * (ky / t_1);
	} else if (th <= 1.35e+15) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (th <= 4.1e+151) {
		tmp = sin(th) / (t_1 / ky);
	} else {
		tmp = sin(th) / (hypot(sin(ky), kx) / sin(ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(ky, Math.sin(kx));
	double tmp;
	if (th <= -2.3e-5) {
		tmp = Math.sin(th) * (ky / t_1);
	} else if (th <= 1.35e+15) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else if (th <= 4.1e+151) {
		tmp = Math.sin(th) / (t_1 / ky);
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(ky), kx) / Math.sin(ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(ky, math.sin(kx))
	tmp = 0
	if th <= -2.3e-5:
		tmp = math.sin(th) * (ky / t_1)
	elif th <= 1.35e+15:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	elif th <= 4.1e+151:
		tmp = math.sin(th) / (t_1 / ky)
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(ky), kx) / math.sin(ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(ky, sin(kx))
	tmp = 0.0
	if (th <= -2.3e-5)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	elseif (th <= 1.35e+15)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (th <= 4.1e+151)
		tmp = Float64(sin(th) / Float64(t_1 / ky));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), kx) / sin(ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(ky, sin(kx));
	tmp = 0.0;
	if (th <= -2.3e-5)
		tmp = sin(th) * (ky / t_1);
	elseif (th <= 1.35e+15)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	elseif (th <= 4.1e+151)
		tmp = sin(th) / (t_1 / ky);
	else
		tmp = sin(th) / (hypot(sin(ky), kx) / sin(ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, -2.3e-5], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.35e+15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[th, 4.1e+151], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / ky), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if th < -2.3e-5

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. unpow2 92.2%

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

      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 50.2%

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

   5. Taylor expanded in ky around 0 62.1%

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

   if -2.3e-5 < th < 1.35e15

   1. Initial program 93.8%

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

      1. associate-/r/ 93.8%

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

      2. +-commutative 93.8%

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

      3. unpow2 93.8%

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

      4. unpow2 93.8%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 92.3%

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

      1. associate-*r/ 92.4%

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

      2. unpow2 92.4%

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

      3. unpow2 92.4%

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

      4. hypot-def 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. hypot-def 92.4%

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

      7. unpow2 92.4%

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

      8. unpow2 92.4%

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

      9. +-commutative 92.4%

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

      10. unpow2 92.4%

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

      11. unpow2 92.4%

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

      12. hypot-def 98.3%

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

   6. Simplified 98.3%

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

      1. associate-/r/ 98.4%

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

      2. hypot-udef 92.4%

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

      3. +-commutative 92.4%

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

      4. hypot-udef 98.4%

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

   8. Applied egg-rr 98.4%

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

   if 1.35e15 < th < 4.0999999999999998e151

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. unpow2 95.5%

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

      3. unpow2 95.5%

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

      4. hypot-def 99.4%

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

   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 63.5%

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

   5. Taylor expanded in ky around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. clear-num 75.9%

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

      3. un-div-inv 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0999999999999998e151 < th

   1. Initial program 96.5%

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

      1. +-commutative 96.5%

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

      2. unpow2 96.5%

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

      3. unpow2 96.5%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in kx around 0 67.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;th \leq 4.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{\sin ky}}\\ \end{array} \]

Alternative 16: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.001)
   (* (sin ky) (/ (sin th) (hypot (sin ky) kx)))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.001) {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.001) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.001:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.001)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.001)
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.001], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1e-3

   1. Initial program 92.0%

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

      1. associate-*l/ 89.8%

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

      2. associate-*r/ 91.9%

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

      3. +-commutative 91.9%

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

      4. unpow2 91.9%

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

      5. unpow2 91.9%

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

      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. Taylor expanded in kx around 0 69.3%

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

   if 1e-3 < kx

   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 99.5%

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

   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 39.3%

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

      1. add-sqr-sqrt 26.0%

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

      2. sqrt-prod 58.3%

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

      3. rem-sqrt-square 58.3%

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

   6. Applied egg-rr 58.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]

Alternative 17: 57.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 1e-6)
   (* (sin th) (/ ky (hypot ky kx)))
   (* (sin ky) (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 1e-6) {
		tmp = sin(th) * (ky / hypot(ky, kx));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, kx));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= 1e-6:
		tmp = math.sin(th) * (ky / math.hypot(ky, kx))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, kx)));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= 1e-6)
		tmp = sin(th) * (ky / hypot(ky, kx));
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 kx) < 9.99999999999999955e-7

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. unpow2 92.2%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in ky around 0 55.3%

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

   5. Taylor expanded in ky around 0 70.0%

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

   6. Taylor expanded in kx around 0 56.0%

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

   if 9.99999999999999955e-7 < (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. associate-*l/ 99.3%

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

      2. associate-*r/ 99.3%

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

      3. +-commutative 99.3%

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

      4. unpow2 99.3%

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

      5. unpow2 99.3%

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

      6. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in ky around 0 57.0%

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

4. Final simplification 56.2%

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

Alternative 18: 36.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -2.5 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -2.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;ky \leq -1.85 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{-105}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -2.5e-139)
     t_1
     (if (<= ky -2.25e-203)
       (/ ky (/ (sin kx) th))
       (if (<= ky -1.85e-230)
         t_1
         (if (<= ky 8e-105)
           (* (sin th) (/ ky kx))
           (if (<= ky 2.1e+29) (sin th) (if (<= ky 6e+175) t_1 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -2.5e-139) {
		tmp = t_1;
	} else if (ky <= -2.25e-203) {
		tmp = ky / (sin(kx) / th);
	} else if (ky <= -1.85e-230) {
		tmp = t_1;
	} else if (ky <= 8e-105) {
		tmp = sin(th) * (ky / kx);
	} else if (ky <= 2.1e+29) {
		tmp = sin(th);
	} else if (ky <= 6e+175) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-2.5d-139)) then
        tmp = t_1
    else if (ky <= (-2.25d-203)) then
        tmp = ky / (sin(kx) / th)
    else if (ky <= (-1.85d-230)) then
        tmp = t_1
    else if (ky <= 8d-105) then
        tmp = sin(th) * (ky / kx)
    else if (ky <= 2.1d+29) then
        tmp = sin(th)
    else if (ky <= 6d+175) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -2.5e-139) {
		tmp = t_1;
	} else if (ky <= -2.25e-203) {
		tmp = ky / (Math.sin(kx) / th);
	} else if (ky <= -1.85e-230) {
		tmp = t_1;
	} else if (ky <= 8e-105) {
		tmp = Math.sin(th) * (ky / kx);
	} else if (ky <= 2.1e+29) {
		tmp = Math.sin(th);
	} else if (ky <= 6e+175) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -2.5e-139:
		tmp = t_1
	elif ky <= -2.25e-203:
		tmp = ky / (math.sin(kx) / th)
	elif ky <= -1.85e-230:
		tmp = t_1
	elif ky <= 8e-105:
		tmp = math.sin(th) * (ky / kx)
	elif ky <= 2.1e+29:
		tmp = math.sin(th)
	elif ky <= 6e+175:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -2.5e-139)
		tmp = t_1;
	elseif (ky <= -2.25e-203)
		tmp = Float64(ky / Float64(sin(kx) / th));
	elseif (ky <= -1.85e-230)
		tmp = t_1;
	elseif (ky <= 8e-105)
		tmp = Float64(sin(th) * Float64(ky / kx));
	elseif (ky <= 2.1e+29)
		tmp = sin(th);
	elseif (ky <= 6e+175)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -2.5e-139)
		tmp = t_1;
	elseif (ky <= -2.25e-203)
		tmp = ky / (sin(kx) / th);
	elseif (ky <= -1.85e-230)
		tmp = t_1;
	elseif (ky <= 8e-105)
		tmp = sin(th) * (ky / kx);
	elseif (ky <= 2.1e+29)
		tmp = sin(th);
	elseif (ky <= 6e+175)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -2.5e-139], t$95$1, If[LessEqual[ky, -2.25e-203], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, -1.85e-230], t$95$1, If[LessEqual[ky, 8e-105], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.1e+29], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 6e+175], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -2.50000000000000017e-139 or -2.2500000000000001e-203 < ky < -1.84999999999999991e-230 or 2.1000000000000002e29 < ky < 6.0000000000000003e175

   1. Initial program 95.6%

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

      1. +-commutative 95.6%

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

      2. unpow2 95.6%

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

      3. unpow2 95.6%

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

      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 30.0%

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

   5. Taylor expanded in ky around 0 46.0%

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

   6. Taylor expanded in ky around -inf 40.5%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 40.5%

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

   8. Simplified 40.5%

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

   if -2.50000000000000017e-139 < ky < -2.2500000000000001e-203

   1. Initial program 92.0%

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

      1. +-commutative 92.0%

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

      2. unpow2 92.0%

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

      3. unpow2 92.0%

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

      4. hypot-def 99.5%

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

   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 56.0%

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

   5. Taylor expanded in th around 0 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-/l* 41.1%

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

   7. Simplified 41.1%

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

   if -1.84999999999999991e-230 < ky < 7.99999999999999972e-105

   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 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 99.7%

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

   5. Taylor expanded in ky around 0 99.7%

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

   6. Taylor expanded in kx around 0 71.1%

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

   7. Taylor expanded in ky around 0 46.1%

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

   if 7.99999999999999972e-105 < ky < 2.1000000000000002e29 or 6.0000000000000003e175 < ky

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 40.4%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -2.5 \cdot 10^{-139}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -2.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;ky \leq -1.85 \cdot 10^{-230}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{-105}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19: 30.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -1.62 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.95 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -1.62e-305)
     t_1
     (if (<= ky 1.95e-105)
       (sqrt (* th th))
       (if (<= ky 2.1e+29) (sin th) (if (<= ky 6e+175) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -1.62e-305) {
		tmp = t_1;
	} else if (ky <= 1.95e-105) {
		tmp = sqrt((th * th));
	} else if (ky <= 2.1e+29) {
		tmp = sin(th);
	} else if (ky <= 6e+175) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-1.62d-305)) then
        tmp = t_1
    else if (ky <= 1.95d-105) then
        tmp = sqrt((th * th))
    else if (ky <= 2.1d+29) then
        tmp = sin(th)
    else if (ky <= 6d+175) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -1.62e-305) {
		tmp = t_1;
	} else if (ky <= 1.95e-105) {
		tmp = Math.sqrt((th * th));
	} else if (ky <= 2.1e+29) {
		tmp = Math.sin(th);
	} else if (ky <= 6e+175) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -1.62e-305:
		tmp = t_1
	elif ky <= 1.95e-105:
		tmp = math.sqrt((th * th))
	elif ky <= 2.1e+29:
		tmp = math.sin(th)
	elif ky <= 6e+175:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -1.62e-305)
		tmp = t_1;
	elseif (ky <= 1.95e-105)
		tmp = sqrt(Float64(th * th));
	elseif (ky <= 2.1e+29)
		tmp = sin(th);
	elseif (ky <= 6e+175)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -1.62e-305)
		tmp = t_1;
	elseif (ky <= 1.95e-105)
		tmp = sqrt((th * th));
	elseif (ky <= 2.1e+29)
		tmp = sin(th);
	elseif (ky <= 6e+175)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -1.62e-305], t$95$1, If[LessEqual[ky, 1.95e-105], N[Sqrt[N[(th * th), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 2.1e+29], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 6e+175], t$95$1, N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.61999999999999999e-305 or 2.1000000000000002e29 < ky < 6.0000000000000003e175

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. unpow2 94.4%

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

      3. unpow2 94.4%

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

      4. hypot-def 99.7%

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

   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 43.1%

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

   5. Taylor expanded in ky around 0 56.1%

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

   6. Taylor expanded in ky around -inf 35.7%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 35.7%

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

   8. Simplified 35.7%

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

   if -1.61999999999999999e-305 < ky < 1.95e-105

   1. Initial program 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. unpow2 85.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r/ 99.7%

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

      2. div-inv 99.4%

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

      3. associate-/r* 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in kx around 0 9.9%

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

   7. Taylor expanded in th around 0 8.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th}}} \]
   8. Step-by-step derivation

      1. remove-double-div 8.7%

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

      2. add-sqr-sqrt 4.6%

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

      3. sqrt-unprod 29.6%

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

   9. Applied egg-rr 29.6%

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

   if 1.95e-105 < ky < 2.1000000000000002e29 or 6.0000000000000003e175 < ky

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 39.9%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.62 \cdot 10^{-305}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 1.95 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 20: 36.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -1.85 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-104}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -1.85e-230)
     t_1
     (if (<= ky 1.8e-104)
       (* (sin th) (/ ky kx))
       (if (<= ky 2.1e+29) (sin th) (if (<= ky 6e+175) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -1.85e-230) {
		tmp = t_1;
	} else if (ky <= 1.8e-104) {
		tmp = sin(th) * (ky / kx);
	} else if (ky <= 2.1e+29) {
		tmp = sin(th);
	} else if (ky <= 6e+175) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-1.85d-230)) then
        tmp = t_1
    else if (ky <= 1.8d-104) then
        tmp = sin(th) * (ky / kx)
    else if (ky <= 2.1d+29) then
        tmp = sin(th)
    else if (ky <= 6d+175) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -1.85e-230) {
		tmp = t_1;
	} else if (ky <= 1.8e-104) {
		tmp = Math.sin(th) * (ky / kx);
	} else if (ky <= 2.1e+29) {
		tmp = Math.sin(th);
	} else if (ky <= 6e+175) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -1.85e-230:
		tmp = t_1
	elif ky <= 1.8e-104:
		tmp = math.sin(th) * (ky / kx)
	elif ky <= 2.1e+29:
		tmp = math.sin(th)
	elif ky <= 6e+175:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -1.85e-230)
		tmp = t_1;
	elseif (ky <= 1.8e-104)
		tmp = Float64(sin(th) * Float64(ky / kx));
	elseif (ky <= 2.1e+29)
		tmp = sin(th);
	elseif (ky <= 6e+175)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -1.85e-230)
		tmp = t_1;
	elseif (ky <= 1.8e-104)
		tmp = sin(th) * (ky / kx);
	elseif (ky <= 2.1e+29)
		tmp = sin(th);
	elseif (ky <= 6e+175)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -1.85e-230], t$95$1, If[LessEqual[ky, 1.8e-104], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.1e+29], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 6e+175], t$95$1, N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.84999999999999991e-230 or 2.1000000000000002e29 < ky < 6.0000000000000003e175

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. unpow2 95.3%

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

      3. unpow2 95.3%

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

      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 ky around 0 36.8%

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

   5. Taylor expanded in ky around 0 51.2%

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

   6. Taylor expanded in ky around -inf 38.5%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 38.5%

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

   8. Simplified 38.5%

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

   if -1.84999999999999991e-230 < ky < 1.7999999999999999e-104

   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 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 99.7%

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

   5. Taylor expanded in ky around 0 99.7%

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

   6. Taylor expanded in kx around 0 71.1%

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

   7. Taylor expanded in ky around 0 46.1%

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

   if 1.7999999999999999e-104 < ky < 2.1000000000000002e29 or 6.0000000000000003e175 < ky

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 40.4%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.85 \cdot 10^{-230}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-104}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 21: 30.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -3 \cdot 10^{-308} \lor \neg \left(ky \leq 2.1 \cdot 10^{+29}\right) \land ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -3e-308) (and (not (<= ky 2.1e+29)) (<= ky 6e+175)))
   (- (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3e-308) || (!(ky <= 2.1e+29) && (ky <= 6e+175))) {
		tmp = -sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-3d-308)) .or. (.not. (ky <= 2.1d+29)) .and. (ky <= 6d+175)) then
        tmp = -sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3e-308) || (!(ky <= 2.1e+29) && (ky <= 6e+175))) {
		tmp = -Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -3e-308) or (not (ky <= 2.1e+29) and (ky <= 6e+175)):
		tmp = -math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -3e-308) || (!(ky <= 2.1e+29) && (ky <= 6e+175)))
		tmp = Float64(-sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -3e-308) || (~((ky <= 2.1e+29)) && (ky <= 6e+175)))
		tmp = -sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -3e-308], And[N[Not[LessEqual[ky, 2.1e+29]], $MachinePrecision], LessEqual[ky, 6e+175]]], (-N[Sin[th], $MachinePrecision]), N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -3.00000000000000022e-308 or 2.1000000000000002e29 < ky < 6.0000000000000003e175

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. unpow2 94.4%

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

      3. unpow2 94.4%

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

      4. hypot-def 99.7%

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

   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 43.1%

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

   5. Taylor expanded in ky around 0 56.1%

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

   6. Taylor expanded in ky around -inf 35.7%

      \[\leadsto \color{blue}{-1 \cdot \sin th} \]
   7. Step-by-step derivation

      1. neg-mul-1 35.7%

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

   8. Simplified 35.7%

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

   if -3.00000000000000022e-308 < ky < 2.1000000000000002e29 or 6.0000000000000003e175 < ky

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. unpow2 93.2%

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

      3. unpow2 93.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 26.6%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3 \cdot 10^{-308} \lor \neg \left(ky \leq 2.1 \cdot 10^{+29}\right) \land ky \leq 6 \cdot 10^{+175}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 22: 23.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

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

Python

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

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 93.9%

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

   1. +-commutative 93.9%

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

   2. unpow2 93.9%

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

   3. unpow2 93.9%

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

   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 21.0%

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

5. Final simplification 21.0%

   \[\leadsto \sin th \]

Alternative 23: 14.6% accurate, 78.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ 1.0 (+ (/ 1.0 th) (* th 0.16666666666666666))))

C

double code(double kx, double ky, double th) {
	return 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 1.0d0 / ((1.0d0 / th) + (th * 0.16666666666666666d0))
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
}

Python

def code(kx, ky, th):
	return 1.0 / ((1.0 / th) + (th * 0.16666666666666666))

Julia

function code(kx, ky, th)
	return Float64(1.0 / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

   1. +-commutative 93.9%

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

   2. unpow2 93.9%

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

   3. unpow2 93.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

   1. associate-/r/ 99.6%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in kx around 0 21.0%

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

7. Taylor expanded in th around 0 12.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
8. Step-by-step derivation

   1. *-commutative 12.1%

      \[\leadsto \frac{1}{\frac{1}{th} + \color{blue}{th \cdot 0.16666666666666666}} \]

9. Simplified 12.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{th} + th \cdot 0.16666666666666666}} \]

10. Final simplification 12.1%

    \[\leadsto \frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666} \]

Alternative 24: 13.9% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.9%

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

   1. +-commutative 93.9%

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

   2. unpow2 93.9%

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

   3. unpow2 93.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

   1. associate-/r/ 99.6%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in kx around 0 21.0%

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

7. Taylor expanded in th around 0 11.6%

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

8. Final simplification 11.6%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023230 
(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)))