Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 23.1s

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: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. *-commutative 91.9%

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

   2. clear-num 91.9%

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

   3. +-commutative 91.9%

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

   4. unpow2 91.9%

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

   5. unpow2 91.9%

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

   6. hypot-udef 99.7%

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

   7. un-div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.21:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.21)
   (fabs th)
   (if (<= (sin ky) 2e-5)
     (* (sin th) (/ ky (hypot (sin ky) (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.21) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(th) * (ky / hypot(sin(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) <= -0.21) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(th) * (ky / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.21:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(th) * (ky / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.21)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(th) * Float64(ky / hypot(sin(ky), sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.21)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-5)
		tmp = sin(th) * (ky / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.21], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 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) < -0.209999999999999992

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.8%

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

      1. +-commutative 40.8%

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

   6. Simplified 40.8%

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

   7. Taylor expanded in kx around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. lft-mult-inverse 3.1%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.1%

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

      4. add-sqr-sqrt 1.2%

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

      5. sqrt-unprod 10.4%

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

      6. pow2 10.4%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 10.4%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 10.4%

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

       2. rem-sqrt-square 19.9%

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

   11. Simplified 19.9%

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

   if -0.209999999999999992 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 85.5%

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

      1. +-commutative 85.5%

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

      2. unpow2 85.5%

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

      3. unpow2 85.5%

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

      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 90.4%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.21:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.02)
     (* (sin ky) (/ th t_1))
     (if (<= (sin ky) 2e-5) (* (sin th) (/ ky t_1)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = sin(ky) * (th / t_1);
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(th) * (ky / t_1);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.sin(ky) * (th / t_1);
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(th) * (ky / t_1);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.sin(ky) * (th / t_1)
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(th) * (ky / t_1)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = sin(ky) * (th / t_1);
	elseif (sin(ky) <= 2e-5)
		tmp = sin(th) * (ky / t_1);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 39.7%

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

      1. +-commutative 39.7%

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

   6. Simplified 39.7%

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

      1. expm1-log1p-u 38.4%

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

      2. expm1-udef 4.7%

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

      3. sqrt-div 4.7%

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

      4. metadata-eval 4.7%

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

      5. +-commutative 4.7%

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

      6. unpow2 4.7%

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

      7. unpow2 4.7%

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

      8. hypot-udef 4.7%

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

      9. un-div-inv 4.7%

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

   8. Applied egg-rr 4.7%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 39.8%

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

      3. hypot-def 39.8%

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

      4. unpow2 39.8%

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

      5. unpow2 39.8%

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

      6. +-commutative 39.8%

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

      7. unpow2 39.8%

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

      8. unpow2 39.8%

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

      9. hypot-def 39.8%

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

   10. Simplified 39.8%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.0%

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

      1. +-commutative 84.0%

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

      2. unpow2 84.0%

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

      3. unpow2 84.0%

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

      4. hypot-def 99.6%

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

   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 99.0%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.045)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) 2e-5)
     (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.045) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(th) * (sin(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) <= -0.045) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(th) * (Math.sin(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) <= -0.045:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(th) * (math.sin(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) <= -0.045)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(th) * Float64(sin(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) <= -0.045)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 2e-5)
		tmp = sin(th) * (sin(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], -0.045], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / 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) < -0.044999999999999998

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.9%

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

      1. +-commutative 40.9%

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

   6. Simplified 40.9%

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

      1. expm1-log1p-u 39.6%

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

      2. expm1-udef 4.8%

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

      3. sqrt-div 4.8%

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

      4. metadata-eval 4.8%

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

      5. +-commutative 4.8%

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

      6. unpow2 4.8%

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

      7. unpow2 4.8%

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

      8. hypot-udef 4.8%

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

      9. un-div-inv 4.8%

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

   8. Applied egg-rr 4.8%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 41.0%

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

      3. hypot-def 41.0%

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

      4. unpow2 41.0%

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

      5. unpow2 41.0%

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

      6. +-commutative 41.0%

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

      7. unpow2 41.0%

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

      8. unpow2 41.0%

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

      9. hypot-def 41.0%

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

   10. Simplified 41.0%

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

   if -0.044999999999999998 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. unpow2 84.2%

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

      3. unpow2 84.2%

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

      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 98.1%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.045)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) 2e-5)
     (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.045) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(ky) * (sin(th) / 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) <= -0.045) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(ky) * (Math.sin(th) / 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) <= -0.045:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(ky) * (math.sin(th) / 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) <= -0.045)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(ky) * Float64(sin(th) / 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) <= -0.045)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 2e-5)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.045], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / 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) < -0.044999999999999998

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.9%

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

      1. +-commutative 40.9%

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

   6. Simplified 40.9%

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

      1. expm1-log1p-u 39.6%

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

      2. expm1-udef 4.8%

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

      3. sqrt-div 4.8%

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

      4. metadata-eval 4.8%

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

      5. +-commutative 4.8%

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

      6. unpow2 4.8%

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

      7. unpow2 4.8%

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

      8. hypot-udef 4.8%

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

      9. un-div-inv 4.8%

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

   8. Applied egg-rr 4.8%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 41.0%

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

      3. hypot-def 41.0%

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

      4. unpow2 41.0%

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

      5. unpow2 41.0%

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

      6. +-commutative 41.0%

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

      7. unpow2 41.0%

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

      8. unpow2 41.0%

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

      9. hypot-def 41.0%

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

   10. Simplified 41.0%

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

   if -0.044999999999999998 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.2%

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

      1. associate-*l/ 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*l/ 84.3%

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

      4. +-commutative 84.3%

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

      5. unpow2 84.3%

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

      6. unpow2 84.3%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 98.1%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.045)
   (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
   (if (<= (sin ky) 2e-5)
     (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.045) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(ky) * (sin(th) / 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) <= -0.045) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(ky) * (Math.sin(th) / 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) <= -0.045:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(ky) * (math.sin(th) / 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) <= -0.045)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(ky) * Float64(sin(th) / 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) <= -0.045)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (sin(ky) <= 2e-5)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.045], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / 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) < -0.044999999999999998

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-udef 99.8%

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

      6. associate-*l/ 99.8%

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

      7. expm1-log1p-u 99.7%

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

      8. expm1-udef 64.0%

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

   3. Applied egg-rr 63.9%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in th around 0 40.9%

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

      1. associate-*l/ 41.0%

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

      2. unpow2 41.0%

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

      3. unpow2 41.0%

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

      4. hypot-def 41.0%

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

      5. *-lft-identity 41.0%

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

   8. Simplified 41.0%

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

   if -0.044999999999999998 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.2%

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

      1. associate-*l/ 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*l/ 84.3%

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

      4. +-commutative 84.3%

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

      5. unpow2 84.3%

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

      6. unpow2 84.3%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 98.1%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.045)
   (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
   (if (<= (sin ky) 2e-5)
     (/ (sin th) (/ (hypot (sin kx) ky) (sin ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.045) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.045) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.045:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.045)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.045)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (sin(ky) <= 2e-5)
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.045], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.044999999999999998

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-udef 99.8%

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

      6. associate-*l/ 99.8%

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

      7. expm1-log1p-u 99.7%

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

      8. expm1-udef 64.0%

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

   3. Applied egg-rr 63.9%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in th around 0 40.9%

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

      1. associate-*l/ 41.0%

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

      2. unpow2 41.0%

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

      3. unpow2 41.0%

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

      4. hypot-def 41.0%

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

      5. *-lft-identity 41.0%

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

   8. Simplified 41.0%

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

   if -0.044999999999999998 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. clear-num 84.2%

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

      3. +-commutative 84.2%

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

      4. unpow2 84.2%

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

      5. unpow2 84.2%

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

      6. hypot-udef 99.6%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 98.1%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.045)
   (/ (* th (sin ky)) (hypot (sin ky) (sin kx)))
   (if (<= (sin ky) 2e-5)
     (/ (sin th) (/ (hypot (sin kx) ky) (sin ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.045) {
		tmp = (th * sin(ky)) / hypot(sin(ky), sin(kx));
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.045) {
		tmp = (th * Math.sin(ky)) / Math.hypot(Math.sin(ky), Math.sin(kx));
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.045:
		tmp = (th * math.sin(ky)) / math.hypot(math.sin(ky), math.sin(kx))
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.045)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(ky), sin(kx)));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.045)
		tmp = (th * sin(ky)) / hypot(sin(ky), sin(kx));
	elseif (sin(ky) <= 2e-5)
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.045], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.044999999999999998

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.9%

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

      1. +-commutative 40.9%

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

   6. Simplified 40.9%

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

      1. expm1-log1p-u 39.8%

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

      2. expm1-udef 4.9%

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

   8. Applied egg-rr 4.9%

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

      1. expm1-def 40.0%

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

      2. expm1-log1p 41.1%

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

      3. associate-*r/ 41.1%

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

      4. hypot-def 41.0%

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

      5. unpow2 41.0%

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

      6. unpow2 41.0%

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

      7. +-commutative 41.0%

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

      8. unpow2 41.0%

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

      9. unpow2 41.0%

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

      10. hypot-def 41.1%

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

   10. Simplified 41.1%

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

   if -0.044999999999999998 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. clear-num 84.2%

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

      3. +-commutative 84.2%

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

      4. unpow2 84.2%

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

      5. unpow2 84.2%

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

      6. hypot-udef 99.6%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 98.1%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.045)
   (/
    (sin ky)
    (* (hypot (sin kx) (sin ky)) (+ (/ 1.0 th) (* th 0.16666666666666666))))
   (if (<= (sin ky) 2e-5)
     (/ (sin th) (/ (hypot (sin kx) ky) (sin ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.045) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (sin(ky) <= 2e-5) {
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.045) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (Math.sin(ky) <= 2e-5) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.045:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)))
	elif math.sin(ky) <= 2e-5:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.045)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	elseif (sin(ky) <= 2e-5)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.045)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	elseif (sin(ky) <= 2e-5)
		tmp = sin(th) / (hypot(sin(kx), ky) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.045], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-5], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.044999999999999998

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-udef 99.8%

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

      6. associate-*l/ 99.8%

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

      7. expm1-log1p-u 99.7%

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

      8. expm1-udef 64.0%

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

   3. Applied egg-rr 63.9%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in th around 0 41.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right) + \frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   7. Step-by-step derivation

      1. +-commutative 41.5%

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{th} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}} \]

      2. unpow2 41.5%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      3. unpow2 41.5%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      4. hypot-def 41.5%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} + 0.16666666666666666 \cdot \left(th \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \]

      5. associate-*r* 41.5%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \color{blue}{\left(0.16666666666666666 \cdot th\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. unpow2 41.5%

         \[\leadsto \frac{\sin ky}{\frac{1}{th} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right) + \left(0.16666666666666666 \cdot th\right) \cdot \sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      7. unpow2 41.5%

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

      8. hypot-def 41.5%

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

      9. distribute-rgt-out 41.5%

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

   8. Simplified 41.5%

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

   if -0.044999999999999998 < (sin.f64 ky) < 2.00000000000000016e-5

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. clear-num 84.2%

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

      3. +-commutative 84.2%

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

      4. unpow2 84.2%

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

      5. unpow2 84.2%

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

      6. hypot-udef 99.6%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 98.1%

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

   if 2.00000000000000016e-5 < (sin.f64 ky)

   1. Initial program 99.6%

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

   2. Taylor expanded in kx around 0 65.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.045:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 44.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.13:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sin th}{0.5 \cdot \frac{ky}{\sin kx} + \frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.13)
   (fabs th)
   (if (<= (sin ky) 2e-67)
     (/ (sin th) (+ (* 0.5 (/ ky (sin kx))) (/ (sin kx) ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.13) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = sin(th) / ((0.5 * (ky / sin(kx))) + (sin(kx) / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.13d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = sin(th) / ((0.5d0 * (ky / sin(kx))) + (sin(kx) / ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.13) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = Math.sin(th) / ((0.5 * (ky / Math.sin(kx))) + (Math.sin(kx) / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.13:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = math.sin(th) / ((0.5 * (ky / math.sin(kx))) + (math.sin(kx) / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.13)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(sin(th) / Float64(Float64(0.5 * Float64(ky / sin(kx))) + Float64(sin(kx) / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.13)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = sin(th) / ((0.5 * (ky / sin(kx))) + (sin(kx) / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.13], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(N[Sin[th], $MachinePrecision] / N[(N[(0.5 * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.13

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 41.9%

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

      1. +-commutative 41.9%

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

   6. Simplified 41.9%

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

   7. Taylor expanded in kx around 0 3.2%

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

      1. associate-*l* 3.2%

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

      2. lft-mult-inverse 3.2%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.2%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 10.1%

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

      6. pow2 10.1%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 10.1%

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

       2. rem-sqrt-square 18.7%

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

   11. Simplified 18.7%

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

   if -0.13 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. clear-num 82.4%

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

      3. +-commutative 82.4%

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

      4. unpow2 82.4%

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

      5. unpow2 82.4%

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

      6. hypot-udef 99.6%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 93.3%

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

   5. Taylor expanded in ky around 0 55.1%

      \[\leadsto \frac{\sin th}{\color{blue}{0.5 \cdot \frac{ky}{\sin kx} + \frac{\sin kx}{ky}}} \]

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.13:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sin th}{0.5 \cdot \frac{ky}{\sin kx} + \frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 44.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.15)
   (fabs th)
   (if (<= (sin ky) 2e-67) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.15) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.15d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.15) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.15:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.15)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.15)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.15], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.149999999999999994

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 41.6%

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

      1. +-commutative 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in kx around 0 3.2%

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

      1. associate-*l* 3.2%

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

      2. lft-mult-inverse 3.2%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.2%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 10.1%

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

      6. pow2 10.1%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 10.1%

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

       2. rem-sqrt-square 19.0%

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

   11. Simplified 19.0%

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

   if -0.149999999999999994 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.7%

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

   2. Taylor expanded in ky around 0 54.4%

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

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 44.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.15)
   (fabs th)
   (if (<= (sin ky) 2e-67) (* (sin ky) (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.15) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.15d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.15) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.15:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.15)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.15)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.15], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.149999999999999994

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 41.6%

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

      1. +-commutative 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in kx around 0 3.2%

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

      1. associate-*l* 3.2%

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

      2. lft-mult-inverse 3.2%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.2%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 10.1%

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

      6. pow2 10.1%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 10.1%

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

       2. rem-sqrt-square 19.0%

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

   11. Simplified 19.0%

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

   if -0.149999999999999994 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.7%

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

      1. associate-*l/ 77.5%

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

      2. *-commutative 77.5%

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

      3. associate-*l/ 82.7%

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

      4. +-commutative 82.7%

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

      5. unpow2 82.7%

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

      6. unpow2 82.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 54.5%

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

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 44.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.15)
   (fabs th)
   (if (<= (sin ky) 2e-67) (/ (sin th) (/ (sin kx) (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.15) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.15d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.15) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.15:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.15)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.15)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = sin(th) / (sin(kx) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.15], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.149999999999999994

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 41.6%

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

      1. +-commutative 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in kx around 0 3.2%

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

      1. associate-*l* 3.2%

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

      2. lft-mult-inverse 3.2%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.2%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 10.1%

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

      6. pow2 10.1%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 10.1%

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

       2. rem-sqrt-square 19.0%

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

   11. Simplified 19.0%

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

   if -0.149999999999999994 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. clear-num 82.7%

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

      3. +-commutative 82.7%

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

      4. unpow2 82.7%

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

      5. unpow2 82.7%

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

      6. hypot-udef 99.6%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 54.4%

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

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.15:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. +-commutative 91.9%

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

   2. unpow2 91.9%

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

   3. unpow2 91.9%

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

   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%

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

Alternative 15: 44.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.09:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.09)
   (fabs th)
   (if (<= (sin ky) 2e-67) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.09) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.09d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.09) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.09:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.09)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.09)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.09], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.089999999999999997

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.6%

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

      1. +-commutative 40.6%

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

   6. Simplified 40.6%

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

   7. Taylor expanded in kx around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. lft-mult-inverse 3.1%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.1%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 9.8%

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

      6. pow2 9.8%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 9.8%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 9.8%

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

       2. rem-sqrt-square 18.1%

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

   11. Simplified 18.1%

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

   if -0.089999999999999997 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.1%

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

      1. +-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. unpow2 82.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 55.5%

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

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.09:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 44.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.09:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.09)
   (fabs th)
   (if (<= (sin ky) 2e-67) (/ ky (/ (sin kx) (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.09) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = ky / (sin(kx) / 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 (sin(ky) <= (-0.09d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = ky / (sin(kx) / 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 (Math.sin(ky) <= -0.09) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.09:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.09)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.09)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.09], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.089999999999999997

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.6%

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

      1. +-commutative 40.6%

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

   6. Simplified 40.6%

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

   7. Taylor expanded in kx around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. lft-mult-inverse 3.1%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.1%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 9.8%

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

      6. pow2 9.8%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 9.8%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 9.8%

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

       2. rem-sqrt-square 18.1%

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

   11. Simplified 18.1%

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

   if -0.089999999999999997 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.1%

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

   2. Taylor expanded in ky around 0 51.7%

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

      1. associate-/l* 55.5%

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

   4. Simplified 55.5%

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

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.09:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 44.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.09:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.09)
   (fabs th)
   (if (<= (sin ky) 2e-67) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.09) {
		tmp = fabs(th);
	} else if (sin(ky) <= 2e-67) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.09d0)) then
        tmp = abs(th)
    else if (sin(ky) <= 2d-67) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.09) {
		tmp = Math.abs(th);
	} else if (Math.sin(ky) <= 2e-67) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.09:
		tmp = math.fabs(th)
	elif math.sin(ky) <= 2e-67:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.09)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.09)
		tmp = abs(th);
	elseif (sin(ky) <= 2e-67)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.09], N[Abs[th], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-67], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.089999999999999997

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.6%

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

      1. +-commutative 40.6%

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

   6. Simplified 40.6%

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

   7. Taylor expanded in kx around 0 3.1%

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

      1. associate-*l* 3.1%

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

      2. lft-mult-inverse 3.1%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 3.1%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 9.8%

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

      6. pow2 9.8%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 9.8%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 9.8%

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

       2. rem-sqrt-square 18.1%

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

   11. Simplified 18.1%

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

   if -0.089999999999999997 < (sin.f64 ky) < 1.99999999999999989e-67

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. clear-num 82.1%

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

      3. +-commutative 82.1%

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

      4. unpow2 82.1%

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

      5. unpow2 82.1%

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

      6. hypot-udef 99.6%

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

      7. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in ky around 0 55.5%

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

   if 1.99999999999999989e-67 < (sin.f64 ky)

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 63.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.09:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18: 32.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -3.4 \cdot 10^{+15} \lor \neg \left(ky \leq 2.9 \cdot 10^{-97}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -3.4e+15) (not (<= ky 2.9e-97)))
   (sin th)
   (* th (/ ky (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3.4e+15) || !(ky <= 2.9e-97)) {
		tmp = sin(th);
	} else {
		tmp = th * (ky / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-3.4d+15)) .or. (.not. (ky <= 2.9d-97))) then
        tmp = sin(th)
    else
        tmp = th * (ky / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3.4e+15) || !(ky <= 2.9e-97)) {
		tmp = Math.sin(th);
	} else {
		tmp = th * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -3.4e+15) or not (ky <= 2.9e-97):
		tmp = math.sin(th)
	else:
		tmp = th * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -3.4e+15) || !(ky <= 2.9e-97))
		tmp = sin(th);
	else
		tmp = Float64(th * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -3.4e+15) || ~((ky <= 2.9e-97)))
		tmp = sin(th);
	else
		tmp = th * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -3.4e+15], N[Not[LessEqual[ky, 2.9e-97]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -3.4e15 or 2.8999999999999999e-97 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 37.4%

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

   if -3.4e15 < ky < 2.8999999999999999e-97

   1. Initial program 80.4%

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

      1. associate-*l/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*l/ 80.5%

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

      4. +-commutative 80.5%

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

      5. unpow2 80.5%

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

      6. unpow2 80.5%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 44.7%

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

      1. +-commutative 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in ky around 0 36.2%

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

      1. associate-/l* 40.2%

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

   9. Simplified 40.2%

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

       1. associate-/r/ 40.2%

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

   11. Applied egg-rr 40.2%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.4 \cdot 10^{+15} \lor \neg \left(ky \leq 2.9 \cdot 10^{-97}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 19: 32.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -3.4 \cdot 10^{+15} \lor \neg \left(ky \leq 2.65 \cdot 10^{-97}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -3.4e+15) (not (<= ky 2.65e-97)))
   (sin th)
   (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3.4e+15) || !(ky <= 2.65e-97)) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-3.4d+15)) .or. (.not. (ky <= 2.65d-97))) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3.4e+15) || !(ky <= 2.65e-97)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -3.4e+15) or not (ky <= 2.65e-97):
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -3.4e+15) || !(ky <= 2.65e-97))
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -3.4e+15) || ~((ky <= 2.65e-97)))
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -3.4e+15], N[Not[LessEqual[ky, 2.65e-97]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -3.4e15 or 2.64999999999999996e-97 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 37.4%

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

   if -3.4e15 < ky < 2.64999999999999996e-97

   1. Initial program 80.4%

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

      1. associate-*l/ 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*l/ 80.5%

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

      4. +-commutative 80.5%

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

      5. unpow2 80.5%

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

      6. unpow2 80.5%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 44.7%

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

      1. +-commutative 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in ky around 0 36.2%

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

      1. associate-/l* 40.2%

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

   9. Simplified 40.2%

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

       1. clear-num 40.1%

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

       2. associate-/r/ 40.2%

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

       3. clear-num 40.2%

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

   11. Applied egg-rr 40.2%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.4 \cdot 10^{+15} \lor \neg \left(ky \leq 2.65 \cdot 10^{-97}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]

Alternative 20: 31.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -3.4 \cdot 10^{+15} \lor \neg \left(ky \leq 3.4 \cdot 10^{-154}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -3.4e+15) (not (<= ky 3.4e-154))) (sin th) (* ky (/ th kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3.4e+15) || !(ky <= 3.4e-154)) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / kx);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-3.4d+15)) .or. (.not. (ky <= 3.4d-154))) then
        tmp = sin(th)
    else
        tmp = ky * (th / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -3.4e+15) || !(ky <= 3.4e-154)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -3.4e+15) or not (ky <= 3.4e-154):
		tmp = math.sin(th)
	else:
		tmp = ky * (th / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -3.4e+15) || !(ky <= 3.4e-154))
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -3.4e+15) || ~((ky <= 3.4e-154)))
		tmp = sin(th);
	else
		tmp = ky * (th / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -3.4e+15], N[Not[LessEqual[ky, 3.4e-154]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -3.4e15 or 3.3999999999999998e-154 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0 36.8%

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

   if -3.4e15 < ky < 3.3999999999999998e-154

   1. Initial program 78.6%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-*l/ 78.7%

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

      4. +-commutative 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

      7. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 43.6%

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

      1. +-commutative 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in ky around 0 37.2%

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

      1. associate-/l* 41.5%

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

   9. Simplified 41.5%

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

   10. Taylor expanded in kx around 0 31.8%

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

       1. associate-/l* 36.1%

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

   12. Simplified 36.1%

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

       1. div-inv 36.1%

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

       2. clear-num 36.1%

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

       3. *-commutative 36.1%

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

   14. Applied egg-rr 36.1%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.4 \cdot 10^{+15} \lor \neg \left(ky \leq 3.4 \cdot 10^{-154}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \end{array} \]

Alternative 21: 21.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.55 \cdot 10^{-149}:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;ky \leq 4.5 \cdot 10^{-93}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.55e-149) (fabs th) (if (<= ky 4.5e-93) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.55e-149) {
		tmp = fabs(th);
	} else if (ky <= 4.5e-93) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.55d-149)) then
        tmp = abs(th)
    else if (ky <= 4.5d-93) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.55e-149) {
		tmp = Math.abs(th);
	} else if (ky <= 4.5e-93) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.55e-149:
		tmp = math.fabs(th)
	elif ky <= 4.5e-93:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.55e-149)
		tmp = abs(th);
	elseif (ky <= 4.5e-93)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.55e-149)
		tmp = abs(th);
	elseif (ky <= 4.5e-93)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.55e-149], N[Abs[th], $MachinePrecision], If[LessEqual[ky, 4.5e-93], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.54999999999999994e-149

   1. Initial program 99.7%

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

      1. associate-*l/ 96.3%

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

      2. *-commutative 96.3%

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

      3. associate-*l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. unpow2 99.7%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 39.5%

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

      1. +-commutative 39.5%

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

   6. Simplified 39.5%

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

   7. Taylor expanded in kx around 0 11.1%

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

      1. associate-*l* 11.2%

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

      2. lft-mult-inverse 11.2%

         \[\leadsto th \cdot \color{blue}{1} \]

      3. *-rgt-identity 11.2%

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

      4. add-sqr-sqrt 4.9%

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

      5. sqrt-unprod 7.9%

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

      6. pow2 7.9%

         \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   9. Applied egg-rr 7.9%

      \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 7.9%

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

       2. rem-sqrt-square 15.0%

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

   11. Simplified 15.0%

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

   if -1.54999999999999994e-149 < ky < 4.5000000000000002e-93

   1. Initial program 74.3%

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

      1. associate-*l/ 72.4%

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

      2. *-commutative 72.4%

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

      3. associate-*l/ 74.3%

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

      4. +-commutative 74.3%

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

      5. unpow2 74.3%

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

      6. unpow2 74.3%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 49.2%

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

      1. +-commutative 49.2%

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

   6. Simplified 49.2%

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

   7. Taylor expanded in ky around 0 44.3%

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

      1. associate-/l* 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in kx around 0 38.8%

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

       1. associate-/l* 44.0%

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

   12. Simplified 44.0%

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

       1. associate-/r/ 44.1%

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

   14. Applied egg-rr 44.1%

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

   if 4.5000000000000002e-93 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*l/ 99.5%

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

      4. +-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 50.2%

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

      1. +-commutative 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in kx around 0 23.7%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.55 \cdot 10^{-149}:\\ \;\;\;\;\left|th\right|\\ \mathbf{elif}\;ky \leq 4.5 \cdot 10^{-93}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 22: 21.6% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-94}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.3e-50) th (if (<= ky 1.8e-94) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.3e-50) {
		tmp = th;
	} else if (ky <= 1.8e-94) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.3d-50)) then
        tmp = th
    else if (ky <= 1.8d-94) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.3e-50) {
		tmp = th;
	} else if (ky <= 1.8e-94) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.3e-50:
		tmp = th
	elif ky <= 1.8e-94:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.3e-50)
		tmp = th;
	elseif (ky <= 1.8e-94)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.3e-50)
		tmp = th;
	elseif (ky <= 1.8e-94)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.3e-50], th, If[LessEqual[ky, 1.8e-94], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.3000000000000001e-50 or 1.8e-94 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 45.3%

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

      1. +-commutative 45.3%

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

   6. Simplified 45.3%

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

   7. Taylor expanded in kx around 0 19.3%

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

   if -1.3000000000000001e-50 < ky < 1.8e-94

   1. Initial program 78.8%

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

      1. associate-*l/ 74.3%

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

      2. *-commutative 74.3%

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

      3. associate-*l/ 78.8%

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

      4. +-commutative 78.8%

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

      5. unpow2 78.8%

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

      6. unpow2 78.8%

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

      7. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 48.2%

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

      1. +-commutative 48.2%

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

   6. Simplified 48.2%

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

   7. Taylor expanded in ky around 0 39.0%

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

      1. associate-/l* 43.2%

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

   9. Simplified 43.2%

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

   10. Taylor expanded in kx around 0 32.7%

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

       1. associate-/l* 37.0%

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

   12. Simplified 37.0%

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

       1. associate-/r/ 37.1%

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

   14. Applied egg-rr 37.1%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-94}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 23: 21.6% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.3e-50) th (if (<= ky 1.1e-93) (* ky (/ th kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.3e-50) {
		tmp = th;
	} else if (ky <= 1.1e-93) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.3d-50)) then
        tmp = th
    else if (ky <= 1.1d-93) then
        tmp = ky * (th / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.3e-50) {
		tmp = th;
	} else if (ky <= 1.1e-93) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.3e-50:
		tmp = th
	elif ky <= 1.1e-93:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.3e-50)
		tmp = th;
	elseif (ky <= 1.1e-93)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.3e-50)
		tmp = th;
	elseif (ky <= 1.1e-93)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.3e-50], th, If[LessEqual[ky, 1.1e-93], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.3000000000000001e-50 or 1.09999999999999998e-93 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 45.3%

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

      1. +-commutative 45.3%

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

   6. Simplified 45.3%

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

   7. Taylor expanded in kx around 0 19.3%

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

   if -1.3000000000000001e-50 < ky < 1.09999999999999998e-93

   1. Initial program 78.8%

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

      1. associate-*l/ 74.3%

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

      2. *-commutative 74.3%

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

      3. associate-*l/ 78.8%

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

      4. +-commutative 78.8%

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

      5. unpow2 78.8%

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

      6. unpow2 78.8%

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

      7. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 48.2%

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

      1. +-commutative 48.2%

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

   6. Simplified 48.2%

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

   7. Taylor expanded in ky around 0 39.0%

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

      1. associate-/l* 43.2%

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

   9. Simplified 43.2%

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

   10. Taylor expanded in kx around 0 32.7%

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

       1. associate-/l* 37.0%

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

   12. Simplified 37.0%

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

       1. div-inv 37.0%

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

       2. clear-num 37.0%

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

       3. *-commutative 37.0%

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

   14. Applied egg-rr 37.0%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 24: 13.2% 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 91.9%

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

   1. associate-*l/ 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*l/ 91.8%

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

   4. +-commutative 91.8%

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

   5. unpow2 91.8%

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

   6. unpow2 91.8%

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

   7. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 46.4%

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

   1. +-commutative 46.4%

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

6. Simplified 46.4%

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

7. Taylor expanded in kx around 0 14.1%

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

8. Final simplification 14.1%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023305 
(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)))