Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 21.3s

Alternatives: 15

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% 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 95.9%

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

   1. +-commutative 95.9%

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

   2. unpow2 95.9%

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

   3. unpow2 95.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.1)
   (- (sin th))
   (if (<= (sin ky) 1e-16) (* (sin th) (/ ky (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.1) {
		tmp = -sin(th);
	} else if (sin(ky) <= 1e-16) {
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.1) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 1e-16) {
		tmp = Math.sin(th) * (ky / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.1:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 1e-16:
		tmp = math.sin(th) * (ky / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.1)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 1e-16)
		tmp = Float64(sin(th) * Float64(ky / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.1)
		tmp = -sin(th);
	elseif (sin(ky) <= 1e-16)
		tmp = sin(th) * (ky / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.1], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-16], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.10000000000000001

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 5.6%

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

   5. Taylor expanded in ky around -inf 4.4%

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

      1. mul-1-neg 4.4%

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

      2. associate-/l* 4.4%

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

      3. distribute-neg-frac 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in ky around 0 66.5%

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

      1. neg-mul-1 66.5%

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

   10. Simplified 66.5%

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

   if -0.10000000000000001 < (sin.f64 ky) < 9.9999999999999998e-17

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. unpow2 91.3%

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

      3. unpow2 91.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 98.6%

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

   5. Taylor expanded in ky around 0 98.9%

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

   if 9.9999999999999998e-17 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 66.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-16}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 55.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-168}:\\ \;\;\;\;\frac{\frac{ky}{\sin kx}}{\frac{1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-29)
   (- (sin th))
   (if (<= (sin ky) 1e-168)
     (/ (/ ky (sin kx)) (/ 1.0 (sin th)))
     (/ (* (sin ky) (sin th)) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-29) {
		tmp = -sin(th);
	} else if (sin(ky) <= 1e-168) {
		tmp = (ky / sin(kx)) / (1.0 / sin(th));
	} else {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-29)) then
        tmp = -sin(th)
    else if (sin(ky) <= 1d-168) then
        tmp = (ky / sin(kx)) / (1.0d0 / sin(th))
    else
        tmp = (sin(ky) * sin(th)) / sin(ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-29) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 1e-168) {
		tmp = (ky / Math.sin(kx)) / (1.0 / Math.sin(th));
	} else {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-29:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 1e-168:
		tmp = (ky / math.sin(kx)) / (1.0 / math.sin(th))
	else:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-29)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 1e-168)
		tmp = Float64(Float64(ky / sin(kx)) / Float64(1.0 / sin(th)));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-29)
		tmp = -sin(th);
	elseif (sin(ky) <= 1e-168)
		tmp = (ky / sin(kx)) / (1.0 / sin(th));
	else
		tmp = (sin(ky) * sin(th)) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-29], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-168], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.99999999999999943e-30

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 12.7%

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

   5. Taylor expanded in ky around -inf 9.2%

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

      1. mul-1-neg 9.2%

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

      2. associate-/l* 9.2%

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

      3. distribute-neg-frac 9.2%

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

   7. Simplified 9.2%

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

   8. Taylor expanded in ky around 0 66.1%

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

      1. neg-mul-1 66.1%

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

   10. Simplified 66.1%

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

   if -9.99999999999999943e-30 < (sin.f64 ky) < 1e-168

   1. Initial program 87.2%

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

      1. +-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. unpow2 87.2%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 53.5%

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

   if 1e-168 < (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. associate-*l/ 98.7%

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

      2. +-commutative 98.7%

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

      3. unpow2 98.7%

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

      4. unpow2 98.7%

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

      5. hypot-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in kx around 0 62.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-168}:\\ \;\;\;\;\frac{\frac{ky}{\sin kx}}{\frac{1}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \]

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 95.9%

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

   1. associate-*l/ 94.4%

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

   2. associate-*r/ 95.8%

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

   3. +-commutative 95.8%

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

   4. unpow2 95.8%

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

   5. unpow2 95.8%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 55.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-29)
   (- (sin th))
   (if (<= (sin ky) 5e-136) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-29) {
		tmp = -sin(th);
	} else if (sin(ky) <= 5e-136) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-29)) then
        tmp = -sin(th)
    else if (sin(ky) <= 5d-136) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-29) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 5e-136) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-29:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 5e-136:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-29)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 5e-136)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-29)
		tmp = -sin(th);
	elseif (sin(ky) <= 5e-136)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-29], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-136], 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) < -9.99999999999999943e-30

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 12.7%

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

   5. Taylor expanded in ky around -inf 9.2%

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

      1. mul-1-neg 9.2%

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

      2. associate-/l* 9.2%

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

      3. distribute-neg-frac 9.2%

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

   7. Simplified 9.2%

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

   8. Taylor expanded in ky around 0 66.1%

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

      1. neg-mul-1 66.1%

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

   10. Simplified 66.1%

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

   if -9.99999999999999943e-30 < (sin.f64 ky) < 5.0000000000000002e-136

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

      2. unpow2 88.1%

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

      3. unpow2 88.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 53.3%

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

   if 5.0000000000000002e-136 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 62.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 55.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-29)
   (- (sin th))
   (if (<= (sin ky) 5e-136) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-29) {
		tmp = -sin(th);
	} else if (sin(ky) <= 5e-136) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-29)) then
        tmp = -sin(th)
    else if (sin(ky) <= 5d-136) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-29) {
		tmp = -Math.sin(th);
	} else if (Math.sin(ky) <= 5e-136) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-29:
		tmp = -math.sin(th)
	elif math.sin(ky) <= 5e-136:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-29)
		tmp = Float64(-sin(th));
	elseif (sin(ky) <= 5e-136)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-29)
		tmp = -sin(th);
	elseif (sin(ky) <= 5e-136)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-29], (-N[Sin[th], $MachinePrecision]), If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-136], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.99999999999999943e-30

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 12.7%

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

   5. Taylor expanded in ky around -inf 9.2%

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

      1. mul-1-neg 9.2%

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

      2. associate-/l* 9.2%

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

      3. distribute-neg-frac 9.2%

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

   7. Simplified 9.2%

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

   8. Taylor expanded in ky around 0 66.1%

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

      1. neg-mul-1 66.1%

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

   10. Simplified 66.1%

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

   if -9.99999999999999943e-30 < (sin.f64 ky) < 5.0000000000000002e-136

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

      2. unpow2 88.1%

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

      3. unpow2 88.1%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 52.4%

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

      1. associate-/l* 53.4%

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

   6. Simplified 53.4%

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

   if 5.0000000000000002e-136 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 62.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 37.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.9 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{ky}{kx}}{\frac{1}{\sin th}}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -1.1e+126)
     t_1
     (if (<= ky -1800.0)
       (sin th)
       (if (<= ky -1.9e-135)
         t_1
         (if (<= ky 1.8e-143)
           (/ (/ ky kx) (/ 1.0 (sin th)))
           (if (<= ky 8e+58)
             (sin th)
             (if (<= ky 1.05e+138) t_1 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = sin(th);
	} else if (ky <= -1.9e-135) {
		tmp = t_1;
	} else if (ky <= 1.8e-143) {
		tmp = (ky / kx) / (1.0 / sin(th));
	} else if (ky <= 8e+58) {
		tmp = sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-1.1d+126)) then
        tmp = t_1
    else if (ky <= (-1800.0d0)) then
        tmp = sin(th)
    else if (ky <= (-1.9d-135)) then
        tmp = t_1
    else if (ky <= 1.8d-143) then
        tmp = (ky / kx) / (1.0d0 / sin(th))
    else if (ky <= 8d+58) then
        tmp = sin(th)
    else if (ky <= 1.05d+138) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = Math.sin(th);
	} else if (ky <= -1.9e-135) {
		tmp = t_1;
	} else if (ky <= 1.8e-143) {
		tmp = (ky / kx) / (1.0 / Math.sin(th));
	} else if (ky <= 8e+58) {
		tmp = Math.sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -1.1e+126:
		tmp = t_1
	elif ky <= -1800.0:
		tmp = math.sin(th)
	elif ky <= -1.9e-135:
		tmp = t_1
	elif ky <= 1.8e-143:
		tmp = (ky / kx) / (1.0 / math.sin(th))
	elif ky <= 8e+58:
		tmp = math.sin(th)
	elif ky <= 1.05e+138:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -1.9e-135)
		tmp = t_1;
	elseif (ky <= 1.8e-143)
		tmp = Float64(Float64(ky / kx) / Float64(1.0 / sin(th)));
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -1.9e-135)
		tmp = t_1;
	elseif (ky <= 1.8e-143)
		tmp = (ky / kx) / (1.0 / sin(th));
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -1.1e+126], t$95$1, If[LessEqual[ky, -1800.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -1.9e-135], t$95$1, If[LessEqual[ky, 1.8e-143], N[(N[(ky / kx), $MachinePrecision] / N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 8e+58], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.05e+138], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.09999999999999999e126 or -1800 < ky < -1.9000000000000001e-135 or 7.99999999999999955e58 < ky < 1.05000000000000003e138

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 33.2%

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

   5. Taylor expanded in ky around -inf 16.5%

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

      1. mul-1-neg 16.5%

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

      2. associate-/l* 16.6%

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

      3. distribute-neg-frac 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in ky around 0 46.8%

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

      1. neg-mul-1 46.8%

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

   10. Simplified 46.8%

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

   if -1.09999999999999999e126 < ky < -1800 or 1.7999999999999999e-143 < ky < 7.99999999999999955e58 or 1.05000000000000003e138 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 45.2%

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

   if -1.9000000000000001e-135 < ky < 1.7999999999999999e-143

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 62.4%

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

   7. Taylor expanded in kx around 0 47.2%

      \[\leadsto \frac{\color{blue}{\frac{ky}{kx}}}{\frac{1}{\sin th}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.9 \cdot 10^{-135}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{ky}{kx}}{\frac{1}{\sin th}}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 32.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.8 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -1.1e+126)
     t_1
     (if (<= ky -1800.0)
       (sin th)
       (if (<= ky -2.8e-152)
         t_1
         (if (<= ky 3.6e-195)
           (sqrt (* th th))
           (if (<= ky 8e+58)
             (sin th)
             (if (<= ky 1.05e+138) t_1 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = sin(th);
	} else if (ky <= -2.8e-152) {
		tmp = t_1;
	} else if (ky <= 3.6e-195) {
		tmp = sqrt((th * th));
	} else if (ky <= 8e+58) {
		tmp = sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-1.1d+126)) then
        tmp = t_1
    else if (ky <= (-1800.0d0)) then
        tmp = sin(th)
    else if (ky <= (-2.8d-152)) then
        tmp = t_1
    else if (ky <= 3.6d-195) then
        tmp = sqrt((th * th))
    else if (ky <= 8d+58) then
        tmp = sin(th)
    else if (ky <= 1.05d+138) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = Math.sin(th);
	} else if (ky <= -2.8e-152) {
		tmp = t_1;
	} else if (ky <= 3.6e-195) {
		tmp = Math.sqrt((th * th));
	} else if (ky <= 8e+58) {
		tmp = Math.sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -1.1e+126:
		tmp = t_1
	elif ky <= -1800.0:
		tmp = math.sin(th)
	elif ky <= -2.8e-152:
		tmp = t_1
	elif ky <= 3.6e-195:
		tmp = math.sqrt((th * th))
	elif ky <= 8e+58:
		tmp = math.sin(th)
	elif ky <= 1.05e+138:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -2.8e-152)
		tmp = t_1;
	elseif (ky <= 3.6e-195)
		tmp = sqrt(Float64(th * th));
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -2.8e-152)
		tmp = t_1;
	elseif (ky <= 3.6e-195)
		tmp = sqrt((th * th));
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -1.1e+126], t$95$1, If[LessEqual[ky, -1800.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -2.8e-152], t$95$1, If[LessEqual[ky, 3.6e-195], N[Sqrt[N[(th * th), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky, 8e+58], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.05e+138], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.09999999999999999e126 or -1800 < ky < -2.79999999999999984e-152 or 7.99999999999999955e58 < ky < 1.05000000000000003e138

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 37.1%

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

   5. Taylor expanded in ky around -inf 15.9%

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

      1. mul-1-neg 15.9%

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

      2. associate-/l* 15.9%

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

      3. distribute-neg-frac 15.9%

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

   7. Simplified 15.9%

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

   8. Taylor expanded in ky around 0 44.4%

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

      1. neg-mul-1 44.4%

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

   10. Simplified 44.4%

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

   if -1.09999999999999999e126 < ky < -1800 or 3.6e-195 < ky < 7.99999999999999955e58 or 1.05000000000000003e138 < ky

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. unpow2 97.4%

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

      3. unpow2 97.4%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 45.0%

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

   if -2.79999999999999984e-152 < ky < 3.6e-195

   1. Initial program 85.5%

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

      1. +-commutative 85.5%

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

      2. unpow2 85.5%

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

      3. unpow2 85.5%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in kx around 0 7.1%

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

   7. Taylor expanded in th around 0 4.8%

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

      1. remove-double-div 4.8%

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

      2. add-sqr-sqrt 2.8%

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

      3. sqrt-unprod 39.0%

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

   9. Applied egg-rr 39.0%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.8 \cdot 10^{-152}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 35.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.7 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -1.1e+126)
     t_1
     (if (<= ky -1800.0)
       (sin th)
       (if (<= ky -3.7e-148)
         t_1
         (if (<= ky 2.8e-195)
           (/ ky (/ (sin kx) th))
           (if (<= ky 8e+58)
             (sin th)
             (if (<= ky 1.05e+138) t_1 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = sin(th);
	} else if (ky <= -3.7e-148) {
		tmp = t_1;
	} else if (ky <= 2.8e-195) {
		tmp = ky / (sin(kx) / th);
	} else if (ky <= 8e+58) {
		tmp = sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-1.1d+126)) then
        tmp = t_1
    else if (ky <= (-1800.0d0)) then
        tmp = sin(th)
    else if (ky <= (-3.7d-148)) then
        tmp = t_1
    else if (ky <= 2.8d-195) then
        tmp = ky / (sin(kx) / th)
    else if (ky <= 8d+58) then
        tmp = sin(th)
    else if (ky <= 1.05d+138) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = Math.sin(th);
	} else if (ky <= -3.7e-148) {
		tmp = t_1;
	} else if (ky <= 2.8e-195) {
		tmp = ky / (Math.sin(kx) / th);
	} else if (ky <= 8e+58) {
		tmp = Math.sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -1.1e+126:
		tmp = t_1
	elif ky <= -1800.0:
		tmp = math.sin(th)
	elif ky <= -3.7e-148:
		tmp = t_1
	elif ky <= 2.8e-195:
		tmp = ky / (math.sin(kx) / th)
	elif ky <= 8e+58:
		tmp = math.sin(th)
	elif ky <= 1.05e+138:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -3.7e-148)
		tmp = t_1;
	elseif (ky <= 2.8e-195)
		tmp = Float64(ky / Float64(sin(kx) / th));
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -3.7e-148)
		tmp = t_1;
	elseif (ky <= 2.8e-195)
		tmp = ky / (sin(kx) / th);
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -1.1e+126], t$95$1, If[LessEqual[ky, -1800.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -3.7e-148], t$95$1, If[LessEqual[ky, 2.8e-195], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 8e+58], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.05e+138], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.09999999999999999e126 or -1800 < ky < -3.70000000000000034e-148 or 7.99999999999999955e58 < ky < 1.05000000000000003e138

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 34.8%

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

   5. Taylor expanded in ky around -inf 16.3%

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

      1. mul-1-neg 16.3%

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

      2. associate-/l* 16.3%

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

      3. distribute-neg-frac 16.3%

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

   7. Simplified 16.3%

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

   8. Taylor expanded in ky around 0 45.8%

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

      1. neg-mul-1 45.8%

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

   10. Simplified 45.8%

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

   if -1.09999999999999999e126 < ky < -1800 or 2.80000000000000003e-195 < ky < 7.99999999999999955e58 or 1.05000000000000003e138 < ky

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. unpow2 97.4%

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

      3. unpow2 97.4%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 45.0%

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

   if -3.70000000000000034e-148 < ky < 2.80000000000000003e-195

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

      2. unpow2 86.3%

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

      3. unpow2 86.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 50.8%

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

   7. Taylor expanded in ky around 0 42.0%

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

      1. associate-/l* 43.7%

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

   9. Simplified 43.7%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.7 \cdot 10^{-148}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 2.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 37.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin th\\ \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -6 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- (sin th))))
   (if (<= ky -1.1e+126)
     t_1
     (if (<= ky -1800.0)
       (sin th)
       (if (<= ky -6e-136)
         t_1
         (if (<= ky 2.6e-144)
           (/ (sin th) (/ kx ky))
           (if (<= ky 8e+58)
             (sin th)
             (if (<= ky 1.05e+138) t_1 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = sin(th);
	} else if (ky <= -6e-136) {
		tmp = t_1;
	} else if (ky <= 2.6e-144) {
		tmp = sin(th) / (kx / ky);
	} else if (ky <= 8e+58) {
		tmp = sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(th)
    if (ky <= (-1.1d+126)) then
        tmp = t_1
    else if (ky <= (-1800.0d0)) then
        tmp = sin(th)
    else if (ky <= (-6d-136)) then
        tmp = t_1
    else if (ky <= 2.6d-144) then
        tmp = sin(th) / (kx / ky)
    else if (ky <= 8d+58) then
        tmp = sin(th)
    else if (ky <= 1.05d+138) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = -Math.sin(th);
	double tmp;
	if (ky <= -1.1e+126) {
		tmp = t_1;
	} else if (ky <= -1800.0) {
		tmp = Math.sin(th);
	} else if (ky <= -6e-136) {
		tmp = t_1;
	} else if (ky <= 2.6e-144) {
		tmp = Math.sin(th) / (kx / ky);
	} else if (ky <= 8e+58) {
		tmp = Math.sin(th);
	} else if (ky <= 1.05e+138) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = -math.sin(th)
	tmp = 0
	if ky <= -1.1e+126:
		tmp = t_1
	elif ky <= -1800.0:
		tmp = math.sin(th)
	elif ky <= -6e-136:
		tmp = t_1
	elif ky <= 2.6e-144:
		tmp = math.sin(th) / (kx / ky)
	elif ky <= 8e+58:
		tmp = math.sin(th)
	elif ky <= 1.05e+138:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(-sin(th))
	tmp = 0.0
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -6e-136)
		tmp = t_1;
	elseif (ky <= 2.6e-144)
		tmp = Float64(sin(th) / Float64(kx / ky));
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = -sin(th);
	tmp = 0.0;
	if (ky <= -1.1e+126)
		tmp = t_1;
	elseif (ky <= -1800.0)
		tmp = sin(th);
	elseif (ky <= -6e-136)
		tmp = t_1;
	elseif (ky <= 2.6e-144)
		tmp = sin(th) / (kx / ky);
	elseif (ky <= 8e+58)
		tmp = sin(th);
	elseif (ky <= 1.05e+138)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = (-N[Sin[th], $MachinePrecision])}, If[LessEqual[ky, -1.1e+126], t$95$1, If[LessEqual[ky, -1800.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, -6e-136], t$95$1, If[LessEqual[ky, 2.6e-144], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 8e+58], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.05e+138], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ky < -1.09999999999999999e126 or -1800 < ky < -5.9999999999999996e-136 or 7.99999999999999955e58 < ky < 1.05000000000000003e138

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 33.2%

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

   5. Taylor expanded in ky around -inf 16.5%

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

      1. mul-1-neg 16.5%

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

      2. associate-/l* 16.6%

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

      3. distribute-neg-frac 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in ky around 0 46.8%

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

      1. neg-mul-1 46.8%

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

   10. Simplified 46.8%

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

   if -1.09999999999999999e126 < ky < -1800 or 2.6000000000000001e-144 < ky < 7.99999999999999955e58 or 1.05000000000000003e138 < ky

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 45.2%

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

   if -5.9999999999999996e-136 < ky < 2.6000000000000001e-144

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 62.4%

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

   7. Taylor expanded in kx around 0 45.8%

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

      1. associate-/l* 47.2%

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

   9. Simplified 47.2%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq -1800:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -6 \cdot 10^{-136}:\\ \;\;\;\;-\sin th\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{elif}\;ky \leq 8 \cdot 10^{+58}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 30.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126} \lor \neg \left(ky \leq -1800\right) \land \left(ky \leq -5 \cdot 10^{-310} \lor \neg \left(ky \leq 8 \cdot 10^{+58}\right) \land ky \leq 1.05 \cdot 10^{+138}\right):\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -1.1e+126)
         (and (not (<= ky -1800.0))
              (or (<= ky -5e-310)
                  (and (not (<= ky 8e+58)) (<= ky 1.05e+138)))))
   (- (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.1e+126) || (!(ky <= -1800.0) && ((ky <= -5e-310) || (!(ky <= 8e+58) && (ky <= 1.05e+138))))) {
		tmp = -sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-1.1d+126)) .or. (.not. (ky <= (-1800.0d0))) .and. (ky <= (-5d-310)) .or. (.not. (ky <= 8d+58)) .and. (ky <= 1.05d+138)) then
        tmp = -sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.1e+126) || (!(ky <= -1800.0) && ((ky <= -5e-310) || (!(ky <= 8e+58) && (ky <= 1.05e+138))))) {
		tmp = -Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -1.1e+126) or (not (ky <= -1800.0) and ((ky <= -5e-310) or (not (ky <= 8e+58) and (ky <= 1.05e+138)))):
		tmp = -math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -1.1e+126) || (!(ky <= -1800.0) && ((ky <= -5e-310) || (!(ky <= 8e+58) && (ky <= 1.05e+138)))))
		tmp = Float64(-sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -1.1e+126) || (~((ky <= -1800.0)) && ((ky <= -5e-310) || (~((ky <= 8e+58)) && (ky <= 1.05e+138)))))
		tmp = -sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -1.1e+126], And[N[Not[LessEqual[ky, -1800.0]], $MachinePrecision], Or[LessEqual[ky, -5e-310], And[N[Not[LessEqual[ky, 8e+58]], $MachinePrecision], LessEqual[ky, 1.05e+138]]]]], (-N[Sin[th], $MachinePrecision]), N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.09999999999999999e126 or -1800 < ky < -4.999999999999985e-310 or 7.99999999999999955e58 < ky < 1.05000000000000003e138

   1. Initial program 96.2%

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

      1. +-commutative 96.2%

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

      2. unpow2 96.2%

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

      3. unpow2 96.2%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 53.4%

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

   5. Taylor expanded in ky around -inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. associate-/l* 15.7%

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

      3. distribute-neg-frac 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in ky around 0 36.8%

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

      1. neg-mul-1 36.8%

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

   10. Simplified 36.8%

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

   if -1.09999999999999999e126 < ky < -1800 or -4.999999999999985e-310 < ky < 7.99999999999999955e58 or 1.05000000000000003e138 < ky

   1. Initial program 95.6%

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

      1. +-commutative 95.6%

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

      2. unpow2 95.6%

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

      3. unpow2 95.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 40.6%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{+126} \lor \neg \left(ky \leq -1800\right) \land \left(ky \leq -5 \cdot 10^{-310} \lor \neg \left(ky \leq 8 \cdot 10^{+58}\right) \land ky \leq 1.05 \cdot 10^{+138}\right):\\ \;\;\;\;-\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 23.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{-306}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;th \leq 1.4 \cdot 10^{-50}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 9.2e-306) (sin th) (if (<= th 1.4e-50) (- th) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 9.2e-306) {
		tmp = sin(th);
	} else if (th <= 1.4e-50) {
		tmp = -th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 9.2d-306) then
        tmp = sin(th)
    else if (th <= 1.4d-50) then
        tmp = -th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 9.2e-306) {
		tmp = Math.sin(th);
	} else if (th <= 1.4e-50) {
		tmp = -th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 9.2e-306:
		tmp = math.sin(th)
	elif th <= 1.4e-50:
		tmp = -th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 9.2e-306)
		tmp = sin(th);
	elseif (th <= 1.4e-50)
		tmp = Float64(-th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 9.2e-306)
		tmp = sin(th);
	elseif (th <= 1.4e-50)
		tmp = -th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 9.2e-306], N[Sin[th], $MachinePrecision], If[LessEqual[th, 1.4e-50], (-th), N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999956e-306 or 1.3999999999999999e-50 < th

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. unpow2 95.5%

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

      3. unpow2 95.5%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 30.5%

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

   if 9.19999999999999956e-306 < th < 1.3999999999999999e-50

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. unpow2 97.7%

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

      3. unpow2 97.7%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/r/ 99.8%

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

      2. div-inv 99.7%

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

      3. associate-/r* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in kx around 0 15.3%

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

      1. remove-double-div 15.3%

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

      2. remove-double-neg 15.3%

         \[\leadsto \color{blue}{-\left(-\sin th\right)} \]

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 49.5%

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

      5. sqr-neg 49.5%

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

      6. sqrt-unprod 33.2%

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

      7. distribute-rgt-neg-in 33.2%

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

   8. Applied egg-rr 33.2%

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

   9. Taylor expanded in th around 0 33.4%

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

       1. neg-mul-1 33.4%

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

   11. Simplified 33.4%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{-306}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;th \leq 1.4 \cdot 10^{-50}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 17.2% accurate, 64.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.9 \cdot 10^{-238}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5.9e-238)
   (- th)
   (/ 1.0 (+ (/ 1.0 th) (* th 0.16666666666666666)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.9e-238) {
		tmp = -th;
	} else {
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
	}
	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 <= (-5.9d-238)) then
        tmp = -th
    else
        tmp = 1.0d0 / ((1.0d0 / th) + (th * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.9e-238) {
		tmp = -th;
	} else {
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.9e-238:
		tmp = -th
	else:
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.9e-238)
		tmp = Float64(-th);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5.9e-238)
		tmp = -th;
	else
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.9e-238], (-th), N[(1.0 / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.8999999999999998e-238

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. unpow2 97.7%

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

      3. unpow2 97.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 17.6%

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

      1. remove-double-div 17.7%

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

      2. remove-double-neg 17.7%

         \[\leadsto \color{blue}{-\left(-\sin th\right)} \]

      3. add-sqr-sqrt 13.0%

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

      4. sqrt-unprod 30.8%

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

      5. sqr-neg 30.8%

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

      6. sqrt-unprod 16.4%

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

      7. distribute-rgt-neg-in 16.4%

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

   8. Applied egg-rr 16.4%

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

   9. Taylor expanded in th around 0 19.3%

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

       1. neg-mul-1 19.3%

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

   11. Simplified 19.3%

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

   if -5.8999999999999998e-238 < ky

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. unpow2 94.5%

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

      3. unpow2 94.5%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/r/ 99.7%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in kx around 0 34.9%

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

   7. Taylor expanded in th around 0 18.0%

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

      1. *-commutative 18.0%

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

   9. Simplified 18.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th} + th \cdot 0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.9 \cdot 10^{-238}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \end{array} \]

Alternative 14: 16.8% accurate, 174.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.9 \cdot 10^{-238}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= ky -5.9e-238) (- th) th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.9e-238) {
		tmp = -th;
	} 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 <= (-5.9d-238)) then
        tmp = -th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.9e-238) {
		tmp = -th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.9e-238:
		tmp = -th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.9e-238)
		tmp = Float64(-th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5.9e-238)
		tmp = -th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.9e-238], (-th), th]

Derivation

1. Split input into 2 regimes

2. if ky < -5.8999999999999998e-238

   1. Initial program 97.7%

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

      1. +-commutative 97.7%

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

      2. unpow2 97.7%

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

      3. unpow2 97.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in kx around 0 17.6%

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

      1. remove-double-div 17.7%

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

      2. remove-double-neg 17.7%

         \[\leadsto \color{blue}{-\left(-\sin th\right)} \]

      3. add-sqr-sqrt 13.0%

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

      4. sqrt-unprod 30.8%

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

      5. sqr-neg 30.8%

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

      6. sqrt-unprod 16.4%

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

      7. distribute-rgt-neg-in 16.4%

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

   8. Applied egg-rr 16.4%

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

   9. Taylor expanded in th around 0 19.3%

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

       1. neg-mul-1 19.3%

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

   11. Simplified 19.3%

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

   if -5.8999999999999998e-238 < ky

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. unpow2 94.5%

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

      3. unpow2 94.5%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/r/ 99.7%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in kx around 0 34.9%

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

   7. Taylor expanded in th around 0 17.4%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.9 \cdot 10^{-238}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 15: 13.9% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 95.9%

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

   1. +-commutative 95.9%

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

   2. unpow2 95.9%

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

   3. unpow2 95.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

   1. associate-/r/ 99.6%

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

   2. div-inv 99.5%

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

   3. associate-/r* 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in kx around 0 27.6%

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

7. Taylor expanded in th around 0 14.7%

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

8. Final simplification 14.7%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023228 
(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)))