Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 11.8s

Alternatives: 17

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

   1. +-commutative 93.5%

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

   2. unpow2 93.5%

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

   3. unpow2 93.5%

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

   4. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-45}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.05)
   (/ (* (sin ky) th) (hypot (sin ky) kx))
   (if (<= (sin ky) 3e-45)
     (* (sin th) (/ (sin ky) (fabs (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.05) {
		tmp = (sin(ky) * th) / hypot(sin(ky), kx);
	} else if (sin(ky) <= 3e-45) {
		tmp = sin(th) * (sin(ky) / fabs(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) <= -0.05) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), kx);
	} else if (Math.sin(ky) <= 3e-45) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.05:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(ky), kx)
	elif math.sin(ky) <= 3e-45:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.05)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(ky), kx));
	elseif (sin(ky) <= 3e-45)
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.05)
		tmp = (sin(ky) * th) / hypot(sin(ky), kx);
	elseif (sin(ky) <= 3e-45)
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.05], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 3e-45], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 59.3%

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

      1. sqrt-div 59.3%

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

      2. metadata-eval 59.3%

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

      3. +-commutative 59.3%

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

      4. unpow2 59.3%

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

      5. unpow2 59.3%

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

      6. hypot-undefine 59.3%

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

      7. un-div-inv 59.4%

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

      8. *-commutative 59.4%

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

   7. Applied egg-rr 59.4%

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

   8. Taylor expanded in kx around 0 28.5%

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

   if -0.050000000000000003 < (sin.f64 ky) < 3.00000000000000011e-45

   1. Initial program 86.6%

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

   3. Taylor expanded in ky around 0 41.1%

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

      1. add-sqr-sqrt 30.7%

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

      2. sqrt-prod 65.0%

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

      3. rem-sqrt-square 71.2%

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

   5. Applied egg-rr 71.2%

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

   if 3.00000000000000011e-45 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 63.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 3 \cdot 10^{-45}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.04:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.04)
   (/ (* (sin ky) th) (hypot (sin ky) kx))
   (if (<= (sin ky) 2e-82) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.04) {
		tmp = (sin(ky) * th) / hypot(sin(ky), kx);
	} else if (sin(ky) <= 2e-82) {
		tmp = sin(th) * (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) <= -0.04) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), kx);
	} else if (Math.sin(ky) <= 2e-82) {
		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) <= -0.04:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(ky), kx)
	elif math.sin(ky) <= 2e-82:
		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) <= -0.04)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(ky), kx));
	elseif (sin(ky) <= 2e-82)
		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) <= -0.04)
		tmp = (sin(ky) * th) / hypot(sin(ky), kx);
	elseif (sin(ky) <= 2e-82)
		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], -0.04], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-82], 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) < -0.0400000000000000008

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 59.9%

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

      1. sqrt-div 59.9%

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

      2. metadata-eval 59.9%

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

      3. +-commutative 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. hypot-undefine 59.9%

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

      7. un-div-inv 60.0%

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

      8. *-commutative 60.0%

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

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in kx around 0 28.2%

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

   if -0.0400000000000000008 < (sin.f64 ky) < 2e-82

   1. Initial program 85.5%

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

   3. Taylor expanded in ky around 0 40.4%

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

   if 2e-82 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.4%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.6%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.04:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

   1. unpow2 93.5%

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

   2. sqr-neg 93.5%

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

   3. sin-neg 93.5%

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

   4. sin-neg 93.5%

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

   5. unpow2 93.5%

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

   6. associate-*l/ 91.4%

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

   7. associate-/l* 93.4%

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

   8. unpow2 93.4%

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

3. Simplified 99.6%

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

Alternative 5: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (fabs (sin th))
   (if (<= (sin ky) 2e-82) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-82) {
		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) <= (-0.005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-82) 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) <= -0.005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-82) {
		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) <= -0.005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-82:
		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) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-82)
		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) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-82)
		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], -0.005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-82], 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) < -0.0050000000000000001

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 24.9%

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

      3. pow2 24.9%

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

   7. Applied egg-rr 24.9%

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

      1. unpow2 24.9%

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

      2. rem-sqrt-square 35.6%

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

   9. Simplified 35.6%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 2e-82

   1. Initial program 85.3%

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

   3. Taylor expanded in ky around 0 41.1%

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

   if 2e-82 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.4%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-82}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (fabs (sin th))
   (if (<= (sin ky) 2e-82) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-82) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.005d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-82) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-82) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-82:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-82)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-82)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-82], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.9%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 24.9%

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

      3. pow2 24.9%

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

   7. Applied egg-rr 24.9%

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

      1. unpow2 24.9%

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

      2. rem-sqrt-square 35.6%

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

   9. Simplified 35.6%

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

   if -0.0050000000000000001 < (sin.f64 ky) < 2e-82

   1. Initial program 85.3%

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

      1. unpow2 85.3%

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

      2. sqr-neg 85.3%

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

      3. sin-neg 85.3%

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

      4. sin-neg 85.3%

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

      5. unpow2 85.3%

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

      6. associate-*l/ 81.4%

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

      7. associate-/l* 85.2%

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

      8. unpow2 85.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 40.3%

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

      1. associate-/l* 41.1%

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

   7. Simplified 41.1%

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

   if 2e-82 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.4%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 60.6%

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

Alternative 7: 69.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.0033:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.0033)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0033) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0033) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.0033:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.0033)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.0033)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.0033], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.0033

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. unpow2 91.7%

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

      3. unpow2 91.7%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in kx around 0 69.4%

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

   if 0.0033 < kx

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0 39.7%

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

      1. add-sqr-sqrt 28.9%

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

      2. sqrt-prod 57.3%

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

      3. rem-sqrt-square 57.3%

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

   5. Applied egg-rr 57.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.0033:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.035:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.035)
   (* (sin ky) (/ (sin th) (hypot (sin ky) kx)))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.035) {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.035) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.035:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.035)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.035)
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.035], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.035000000000000003

   1. Initial program 91.7%

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

      1. unpow2 91.7%

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

      2. sqr-neg 91.7%

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

      3. sin-neg 91.7%

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

      4. sin-neg 91.7%

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

      5. unpow2 91.7%

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

      6. associate-*l/ 89.0%

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

      7. associate-/l* 91.6%

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

      8. unpow2 91.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 69.3%

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

   if 0.035000000000000003 < kx

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0 39.7%

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

      1. add-sqr-sqrt 28.9%

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

      2. sqrt-prod 57.3%

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

      3. rem-sqrt-square 57.3%

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

   5. Applied egg-rr 57.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.035:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.39:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.39)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (* (sin ky) (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.39) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.39) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.39:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.39)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.39)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.39], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.39000000000000001

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 90.8%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 64.6%

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

   if 0.39000000000000001 < th

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 93.7%

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

      7. associate-/l* 93.6%

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

      8. unpow2 93.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 47.5%

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

Alternative 10: 60.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.41:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.41)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.41) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.41) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.41:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.41)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.41)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.41], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.409999999999999976

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 90.8%

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

      7. associate-/l* 93.4%

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

      8. unpow2 93.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 64.6%

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

   if 0.409999999999999976 < th

   1. Initial program 93.8%

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

   3. Taylor expanded in ky around 0 18.3%

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

      1. add-sqr-sqrt 17.2%

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

      2. sqrt-prod 34.9%

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

      3. rem-sqrt-square 37.3%

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

   5. Applied egg-rr 37.3%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.41:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.9e-68) (sin th) (* (sin th) (/ (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.9e-68) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(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 (kx <= 1.9d-68) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.9e-68) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.9e-68:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.9e-68)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.9e-68)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.9e-68], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.90000000000000019e-68

   1. Initial program 91.3%

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

      1. unpow2 91.3%

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

      2. sqr-neg 91.3%

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

      3. sin-neg 91.3%

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

      4. sin-neg 91.3%

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

      5. unpow2 91.3%

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

      6. associate-*l/ 88.4%

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

      7. associate-/l* 91.2%

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

      8. unpow2 91.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 29.3%

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

   if 1.90000000000000019e-68 < kx

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 39.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.6e-68) (sin th) (* (sin ky) (/ (sin th) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-68) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(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 (kx <= 3.6d-68) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.6e-68) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.6e-68:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.6e-68)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.6e-68)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.6e-68], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.60000000000000007e-68

   1. Initial program 91.3%

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

      1. unpow2 91.3%

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

      2. sqr-neg 91.3%

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

      3. sin-neg 91.3%

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

      4. sin-neg 91.3%

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

      5. unpow2 91.3%

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

      6. associate-*l/ 88.4%

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

      7. associate-/l* 91.2%

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

      8. unpow2 91.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 29.3%

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

   if 3.60000000000000007e-68 < kx

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 39.0%

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

Alternative 13: 27.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|th \cdot \frac{ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.9e-37) (sin th) (fabs (* th (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.9e-37) {
		tmp = sin(th);
	} else {
		tmp = fabs((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 (kx <= 2.9d-37) then
        tmp = sin(th)
    else
        tmp = abs((th * (ky / sin(kx))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.9e-37) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.abs((th * (ky / Math.sin(kx))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 2.9e-37:
		tmp = math.sin(th)
	else:
		tmp = math.fabs((th * (ky / math.sin(kx))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.9e-37)
		tmp = sin(th);
	else
		tmp = abs(Float64(th * Float64(ky / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 2.9e-37)
		tmp = sin(th);
	else
		tmp = abs((th * (ky / sin(kx))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.9e-37], N[Sin[th], $MachinePrecision], N[Abs[N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.90000000000000005e-37

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. sqr-neg 91.4%

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

      3. sin-neg 91.4%

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

      4. sin-neg 91.4%

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

      5. unpow2 91.4%

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

      6. associate-*l/ 88.7%

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

      7. associate-/l* 91.4%

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

      8. unpow2 91.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 29.8%

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

   if 2.90000000000000005e-37 < kx

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 53.4%

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

   6. Taylor expanded in ky around 0 23.6%

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

      1. add-sqr-sqrt 17.6%

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

      2. sqrt-unprod 19.8%

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

      3. pow2 19.8%

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

      4. *-commutative 19.8%

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

      5. associate-/l* 19.8%

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

   8. Applied egg-rr 19.8%

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

      1. unpow2 19.8%

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

      2. rem-sqrt-square 19.8%

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

   10. Simplified 19.8%

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

Alternative 14: 27.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.1e-37) (sin th) (* th (/ ky (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.1e-37) {
		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 (kx <= 1.1d-37) 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 (kx <= 1.1e-37) {
		tmp = Math.sin(th);
	} else {
		tmp = th * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.1e-37:
		tmp = math.sin(th)
	else:
		tmp = th * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.1e-37)
		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 (kx <= 1.1e-37)
		tmp = sin(th);
	else
		tmp = th * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.1e-37], N[Sin[th], $MachinePrecision], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.10000000000000001e-37

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. sqr-neg 91.4%

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

      3. sin-neg 91.4%

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

      4. sin-neg 91.4%

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

      5. unpow2 91.4%

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

      6. associate-*l/ 88.7%

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

      7. associate-/l* 91.4%

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

      8. unpow2 91.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 29.8%

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

   if 1.10000000000000001e-37 < kx

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 53.4%

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

   6. Taylor expanded in ky around 0 26.1%

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

   7. Taylor expanded in ky around 0 23.6%

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

      1. *-commutative 23.6%

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

      2. associate-*r/ 23.6%

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

   9. Simplified 23.6%

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

Alternative 15: 24.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7.4 \cdot 10^{-178}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 7.4e-178) (* ky (/ th kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7.4e-178) {
		tmp = ky * (th / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 7.4d-178) then
        tmp = ky * (th / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7.4e-178) {
		tmp = ky * (th / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 7.4e-178:
		tmp = ky * (th / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 7.4e-178)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 7.4e-178)
		tmp = ky * (th / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 7.4e-178], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 7.40000000000000008e-178

   1. Initial program 90.9%

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

      1. unpow2 90.9%

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

      2. sqr-neg 90.9%

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

      3. sin-neg 90.9%

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

      4. sin-neg 90.9%

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

      5. unpow2 90.9%

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

      6. associate-*l/ 89.3%

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

      7. associate-/l* 90.7%

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

      8. unpow2 90.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 47.8%

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

   6. Taylor expanded in ky around 0 15.0%

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

   7. Taylor expanded in kx around 0 15.0%

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

      1. associate-/l* 15.5%

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

   9. Simplified 15.5%

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

   if 7.40000000000000008e-178 < ky

   1. Initial program 98.3%

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

      1. unpow2 98.3%

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

      2. sqr-neg 98.3%

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

      3. sin-neg 98.3%

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

      4. sin-neg 98.3%

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

      5. unpow2 98.3%

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

      6. associate-*l/ 95.2%

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

      7. associate-/l* 98.3%

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

      8. unpow2 98.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 36.4%

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

Alternative 16: 18.4% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.5e-124) (* ky (/ th kx)) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.5e-124) {
		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 <= 4.5d-124) 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 <= 4.5e-124) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.5e-124:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.5e-124)
		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 <= 4.5e-124)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.5e-124], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if ky < 4.4999999999999996e-124

   1. Initial program 90.5%

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

      1. unpow2 90.5%

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

      2. sqr-neg 90.5%

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

      3. sin-neg 90.5%

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

      4. sin-neg 90.5%

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

      5. unpow2 90.5%

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

      6. associate-*l/ 89.1%

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

      7. associate-/l* 90.4%

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

      8. unpow2 90.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 46.5%

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

   6. Taylor expanded in ky around 0 14.5%

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

   7. Taylor expanded in kx around 0 14.5%

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

      1. associate-/l* 15.0%

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

   9. Simplified 15.0%

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

   if 4.4999999999999996e-124 < ky

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 96.3%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 49.0%

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

   6. Taylor expanded in kx around 0 17.7%

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

Alternative 17: 13.7% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.5%

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

   1. unpow2 93.5%

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

   2. sqr-neg 93.5%

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

   3. sin-neg 93.5%

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

   4. sin-neg 93.5%

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

   5. unpow2 93.5%

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

   6. associate-*l/ 91.4%

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

   7. associate-/l* 93.4%

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

   8. unpow2 93.4%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 47.3%

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

6. Taylor expanded in kx around 0 15.0%

   \[\leadsto \color{blue}{th} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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)))