Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 25.1s

Alternatives: 23

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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. +-commutative 93.4%

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

   2. unpow2 93.4%

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

   3. unpow2 93.4%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 48.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ t_2 := \frac{t_1}{\frac{\sin kx}{ky}}\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-288}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))) (t_2 (/ t_1 (/ (sin kx) ky))))
   (if (<= (sin ky) -1e-37)
     t_1
     (if (<= (sin ky) 2e-288)
       t_2
       (if (<= (sin ky) 1e-104)
         (/ (* ky (sin th)) (sin kx))
         (if (<= (sin ky) 1e-98) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double t_2 = t_1 / (sin(kx) / ky);
	double tmp;
	if (sin(ky) <= -1e-37) {
		tmp = t_1;
	} else if (sin(ky) <= 2e-288) {
		tmp = t_2;
	} else if (sin(ky) <= 1e-104) {
		tmp = (ky * sin(th)) / sin(kx);
	} else if (sin(ky) <= 1e-98) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs(sin(th))
    t_2 = t_1 / (sin(kx) / ky)
    if (sin(ky) <= (-1d-37)) then
        tmp = t_1
    else if (sin(ky) <= 2d-288) then
        tmp = t_2
    else if (sin(ky) <= 1d-104) then
        tmp = (ky * sin(th)) / sin(kx)
    else if (sin(ky) <= 1d-98) then
        tmp = t_2
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double t_2 = t_1 / (Math.sin(kx) / ky);
	double tmp;
	if (Math.sin(ky) <= -1e-37) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 2e-288) {
		tmp = t_2;
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else if (Math.sin(ky) <= 1e-98) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	t_2 = t_1 / (math.sin(kx) / ky)
	tmp = 0
	if math.sin(ky) <= -1e-37:
		tmp = t_1
	elif math.sin(ky) <= 2e-288:
		tmp = t_2
	elif math.sin(ky) <= 1e-104:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	elif math.sin(ky) <= 1e-98:
		tmp = t_2
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	t_2 = Float64(t_1 / Float64(sin(kx) / ky))
	tmp = 0.0
	if (sin(ky) <= -1e-37)
		tmp = t_1;
	elseif (sin(ky) <= 2e-288)
		tmp = t_2;
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	elseif (sin(ky) <= 1e-98)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	t_2 = t_1 / (sin(kx) / ky);
	tmp = 0.0;
	if (sin(ky) <= -1e-37)
		tmp = t_1;
	elseif (sin(ky) <= 2e-288)
		tmp = t_2;
	elseif (sin(ky) <= 1e-104)
		tmp = (ky * sin(th)) / sin(kx);
	elseif (sin(ky) <= 1e-98)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-37], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-288], t$95$2, If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-98], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1.00000000000000007e-37

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.1%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

   6. Applied egg-rr 26.6%

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

      1. unpow2 26.6%

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

      2. rem-sqrt-square 35.7%

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

   8. Simplified 35.7%

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

   if -1.00000000000000007e-37 < (sin.f64 ky) < 2.00000000000000012e-288 or 9.99999999999999927e-105 < (sin.f64 ky) < 9.99999999999999939e-99

   1. Initial program 79.1%

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

      1. +-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. unpow2 79.1%

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

      4. hypot-def 99.7%

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

   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 42.0%

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

      1. associate-/l* 41.7%

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

   6. Simplified 41.7%

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

      1. add-sqr-sqrt 1.7%

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

      2. sqrt-unprod 29.4%

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

      3. pow2 29.4%

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

   8. Applied egg-rr 53.2%

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

      1. unpow2 29.4%

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

      2. rem-sqrt-square 16.4%

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

   10. Simplified 57.3%

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

   if 2.00000000000000012e-288 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. unpow2 92.8%

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

      3. unpow2 92.8%

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

      4. hypot-def 99.7%

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

   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 64.7%

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

   if 9.99999999999999939e-99 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 53.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-288}:\\ \;\;\;\;\frac{\left|\sin th\right|}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-98}:\\ \;\;\;\;\frac{\left|\sin th\right|}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.78:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 10^{-16}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.78)
   (fabs (* (sin th) (/ (sin ky) (sin kx))))
   (if (<= (sin th) -0.02)
     (fabs (sin th))
     (if (<= (sin th) 1e-16)
       (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
       (/ (sin ky) (fabs (/ (sin kx) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.78) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else if (sin(th) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= 1e-16) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.78) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else if (Math.sin(th) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= 1e-16) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.78:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	elif math.sin(th) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= 1e-16:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.78)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	elseif (sin(th) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(th) <= 1e-16)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.78)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	elseif (sin(th) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(th) <= 1e-16)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.78], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-16], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.78000000000000003

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 19.9%

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

      1. add-sqr-sqrt 14.0%

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

      2. sqrt-unprod 21.0%

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

      3. pow2 21.0%

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

      4. *-commutative 21.0%

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

   6. Applied egg-rr 21.0%

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

      1. unpow2 21.0%

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

      2. rem-sqrt-square 31.3%

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

   8. Simplified 31.3%

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

   if -0.78000000000000003 < (sin.f64 th) < -0.0200000000000000004

   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. Taylor expanded in kx around 0 19.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 32.0%

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

      3. pow2 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. unpow2 32.0%

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

      2. rem-sqrt-square 32.0%

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

   8. Simplified 32.0%

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

   if -0.0200000000000000004 < (sin.f64 th) < 9.9999999999999998e-17

   1. Initial program 94.4%

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

      1. associate-*l/ 90.2%

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

      2. associate-*r/ 94.3%

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

      3. +-commutative 94.3%

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

      4. unpow2 94.3%

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

      5. unpow2 94.3%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 99.7%

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

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

   1. Initial program 94.4%

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

      1. associate-/r/ 94.5%

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

      2. +-commutative 94.5%

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

      3. unpow2 94.5%

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

      4. unpow2 94.5%

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

      5. hypot-def 99.4%

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

   3. Simplified 99.4%

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

      1. div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 27.5%

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

      1. add-sqr-sqrt 26.5%

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

      2. sqrt-unprod 44.9%

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

      3. pow2 44.9%

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

      4. un-div-inv 44.9%

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

   8. Applied egg-rr 44.9%

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

      1. unpow2 44.9%

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

      2. rem-sqrt-square 47.1%

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

   10. Simplified 47.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.78:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 10^{-16}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 4: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.78:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 10^{-16}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.78)
   (fabs (* (sin th) (/ (sin ky) (sin kx))))
   (if (<= (sin th) -0.02)
     (fabs (sin th))
     (if (<= (sin th) 1e-16)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (/ (sin ky) (fabs (/ (sin kx) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.78) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else if (sin(th) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(th) <= 1e-16) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(ky) / fabs((sin(kx) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.78) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else if (Math.sin(th) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(th) <= 1e-16) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(kx) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.78:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	elif math.sin(th) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(th) <= 1e-16:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(kx) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.78)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	elseif (sin(th) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(th) <= 1e-16)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(kx) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.78)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	elseif (sin(th) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(th) <= 1e-16)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = sin(ky) / abs((sin(kx) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.78], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-16], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.78000000000000003

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in ky around 0 19.9%

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

      1. add-sqr-sqrt 14.0%

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

      2. sqrt-unprod 21.0%

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

      3. pow2 21.0%

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

      4. *-commutative 21.0%

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

   6. Applied egg-rr 21.0%

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

      1. unpow2 21.0%

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

      2. rem-sqrt-square 31.3%

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

   8. Simplified 31.3%

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

   if -0.78000000000000003 < (sin.f64 th) < -0.0200000000000000004

   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. Taylor expanded in kx around 0 19.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 32.0%

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

      3. pow2 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. unpow2 32.0%

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

      2. rem-sqrt-square 32.0%

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

   8. Simplified 32.0%

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

   if -0.0200000000000000004 < (sin.f64 th) < 9.9999999999999998e-17

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. unpow2 94.4%

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

      3. unpow2 94.4%

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

      4. hypot-def 99.8%

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

   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 th around 0 99.8%

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

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

   1. Initial program 94.4%

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

      1. associate-/r/ 94.5%

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

      2. +-commutative 94.5%

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

      3. unpow2 94.5%

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

      4. unpow2 94.5%

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

      5. hypot-def 99.4%

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

   3. Simplified 99.4%

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

      1. div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in ky around 0 27.5%

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

      1. add-sqr-sqrt 26.5%

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

      2. sqrt-unprod 44.9%

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

      3. pow2 44.9%

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

      4. un-div-inv 44.9%

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

   8. Applied egg-rr 44.9%

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

      1. unpow2 44.9%

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

      2. rem-sqrt-square 47.1%

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

   10. Simplified 47.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.78:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin th \leq 10^{-16}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 5: 53.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ t_2 := \frac{\sin kx}{ky}\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{t_1}{t_2}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin th}{\left|t_2\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))) (t_2 (/ (sin kx) ky)))
   (if (<= (sin ky) -1e-37)
     t_1
     (if (<= (sin ky) -5e-301)
       (/ t_1 t_2)
       (if (<= (sin ky) 2e-16) (/ (sin th) (fabs t_2)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double t_2 = sin(kx) / ky;
	double tmp;
	if (sin(ky) <= -1e-37) {
		tmp = t_1;
	} else if (sin(ky) <= -5e-301) {
		tmp = t_1 / t_2;
	} else if (sin(ky) <= 2e-16) {
		tmp = sin(th) / fabs(t_2);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs(sin(th))
    t_2 = sin(kx) / ky
    if (sin(ky) <= (-1d-37)) then
        tmp = t_1
    else if (sin(ky) <= (-5d-301)) then
        tmp = t_1 / t_2
    else if (sin(ky) <= 2d-16) then
        tmp = sin(th) / abs(t_2)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double t_2 = Math.sin(kx) / ky;
	double tmp;
	if (Math.sin(ky) <= -1e-37) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -5e-301) {
		tmp = t_1 / t_2;
	} else if (Math.sin(ky) <= 2e-16) {
		tmp = Math.sin(th) / Math.abs(t_2);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	t_2 = math.sin(kx) / ky
	tmp = 0
	if math.sin(ky) <= -1e-37:
		tmp = t_1
	elif math.sin(ky) <= -5e-301:
		tmp = t_1 / t_2
	elif math.sin(ky) <= 2e-16:
		tmp = math.sin(th) / math.fabs(t_2)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	t_2 = Float64(sin(kx) / ky)
	tmp = 0.0
	if (sin(ky) <= -1e-37)
		tmp = t_1;
	elseif (sin(ky) <= -5e-301)
		tmp = Float64(t_1 / t_2);
	elseif (sin(ky) <= 2e-16)
		tmp = Float64(sin(th) / abs(t_2));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	t_2 = sin(kx) / ky;
	tmp = 0.0;
	if (sin(ky) <= -1e-37)
		tmp = t_1;
	elseif (sin(ky) <= -5e-301)
		tmp = t_1 / t_2;
	elseif (sin(ky) <= 2e-16)
		tmp = sin(th) / abs(t_2);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-37], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-301], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-16], N[(N[Sin[th], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1.00000000000000007e-37

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.1%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

   6. Applied egg-rr 26.6%

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

      1. unpow2 26.6%

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

      2. rem-sqrt-square 35.7%

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

   8. Simplified 35.7%

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

   if -1.00000000000000007e-37 < (sin.f64 ky) < -5.00000000000000013e-301

   1. Initial program 79.2%

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

      1. +-commutative 79.2%

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

      2. unpow2 79.2%

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

      3. unpow2 79.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 40.2%

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

      1. associate-/l* 39.9%

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

   6. Simplified 39.9%

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

      1. add-sqr-sqrt 1.6%

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

      2. sqrt-unprod 31.7%

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

      3. pow2 31.7%

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

   8. Applied egg-rr 48.6%

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

      1. unpow2 31.7%

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

      2. rem-sqrt-square 18.3%

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

   10. Simplified 53.4%

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

   if -5.00000000000000013e-301 < (sin.f64 ky) < 2e-16

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. unpow2 92.3%

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

      3. unpow2 92.3%

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

      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 51.5%

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

      1. associate-/l* 51.2%

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

   6. Simplified 51.2%

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

      1. add-sqr-sqrt 40.3%

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

      2. sqrt-unprod 59.9%

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

      3. pow2 59.9%

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

   8. Applied egg-rr 59.9%

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

      1. unpow2 59.9%

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

      2. rem-sqrt-square 80.2%

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

   10. Simplified 80.2%

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

   if 2e-16 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 56.3%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{\left|\sin th\right|}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6: 53.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= (sin ky) -1e-37)
     t_1
     (if (<= (sin ky) 5e-304)
       (* (/ (sin ky) (sin kx)) t_1)
       (if (<= (sin ky) 2e-16)
         (/ (sin th) (fabs (/ (sin kx) ky)))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (sin(ky) <= -1e-37) {
		tmp = t_1;
	} else if (sin(ky) <= 5e-304) {
		tmp = (sin(ky) / sin(kx)) * t_1;
	} else if (sin(ky) <= 2e-16) {
		tmp = sin(th) / fabs((sin(kx) / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (sin(ky) <= (-1d-37)) then
        tmp = t_1
    else if (sin(ky) <= 5d-304) then
        tmp = (sin(ky) / sin(kx)) * t_1
    else if (sin(ky) <= 2d-16) then
        tmp = sin(th) / abs((sin(kx) / ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (Math.sin(ky) <= -1e-37) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 5e-304) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * t_1;
	} else if (Math.sin(ky) <= 2e-16) {
		tmp = Math.sin(th) / Math.abs((Math.sin(kx) / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if math.sin(ky) <= -1e-37:
		tmp = t_1
	elif math.sin(ky) <= 5e-304:
		tmp = (math.sin(ky) / math.sin(kx)) * t_1
	elif math.sin(ky) <= 2e-16:
		tmp = math.sin(th) / math.fabs((math.sin(kx) / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (sin(ky) <= -1e-37)
		tmp = t_1;
	elseif (sin(ky) <= 5e-304)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * t_1);
	elseif (sin(ky) <= 2e-16)
		tmp = Float64(sin(th) / abs(Float64(sin(kx) / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (sin(ky) <= -1e-37)
		tmp = t_1;
	elseif (sin(ky) <= 5e-304)
		tmp = (sin(ky) / sin(kx)) * t_1;
	elseif (sin(ky) <= 2e-16)
		tmp = sin(th) / abs((sin(kx) / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-37], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-304], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-16], N[(N[Sin[th], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1.00000000000000007e-37

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.1%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 26.6%

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

      3. pow2 26.6%

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

   6. Applied egg-rr 26.6%

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

      1. unpow2 26.6%

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

      2. rem-sqrt-square 35.7%

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

   8. Simplified 35.7%

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

   if -1.00000000000000007e-37 < (sin.f64 ky) < 4.99999999999999965e-304

   1. Initial program 79.6%

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

      1. +-commutative 79.6%

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

      2. unpow2 79.6%

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

      3. unpow2 79.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 39.3%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 31.2%

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

      3. pow2 31.2%

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

   6. Applied egg-rr 49.5%

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

      1. unpow2 31.2%

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

      2. rem-sqrt-square 18.0%

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

   8. Simplified 54.2%

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

   if 4.99999999999999965e-304 < (sin.f64 ky) < 2e-16

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. unpow2 92.2%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 52.2%

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

      1. associate-/l* 51.9%

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

   6. Simplified 51.9%

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

      1. add-sqr-sqrt 41.0%

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

      2. sqrt-unprod 60.8%

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

      3. pow2 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. unpow2 60.8%

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

      2. rem-sqrt-square 79.9%

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

   10. Simplified 79.9%

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

   if 2e-16 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 56.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 47.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-42)
   (fabs (sin th))
   (if (<= (sin ky) -2e-281)
     (fabs (* (sin th) (/ ky (sin kx))))
     (if (<= (sin ky) 5e-99) (/ (* ky (sin th)) (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-42) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -2e-281) {
		tmp = fabs((sin(th) * (ky / sin(kx))));
	} else if (sin(ky) <= 5e-99) {
		tmp = (ky * sin(th)) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-42)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-2d-281)) then
        tmp = abs((sin(th) * (ky / sin(kx))))
    else if (sin(ky) <= 5d-99) then
        tmp = (ky * sin(th)) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-42) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -2e-281) {
		tmp = Math.abs((Math.sin(th) * (ky / Math.sin(kx))));
	} else if (Math.sin(ky) <= 5e-99) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-42:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -2e-281:
		tmp = math.fabs((math.sin(th) * (ky / math.sin(kx))))
	elif math.sin(ky) <= 5e-99:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-42)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -2e-281)
		tmp = abs(Float64(sin(th) * Float64(ky / sin(kx))));
	elseif (sin(ky) <= 5e-99)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-42)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -2e-281)
		tmp = abs((sin(th) * (ky / sin(kx))));
	elseif (sin(ky) <= 5e-99)
		tmp = (ky * sin(th)) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-42], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-281], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-99], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1.00000000000000004e-42

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.1%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 26.3%

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

      3. pow2 26.3%

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

   6. Applied egg-rr 26.3%

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

      1. unpow2 26.3%

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

      2. rem-sqrt-square 35.2%

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

   8. Simplified 35.2%

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

   if -1.00000000000000004e-42 < (sin.f64 ky) < -2e-281

   1. Initial program 78.5%

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

      1. +-commutative 78.5%

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

      2. unpow2 78.5%

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

      3. unpow2 78.5%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 39.7%

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

      1. associate-/l* 39.5%

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

   6. Simplified 39.5%

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

      1. add-sqr-sqrt 28.9%

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

      2. sqrt-unprod 40.1%

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

      3. pow2 40.1%

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

      4. associate-/r/ 40.1%

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

      5. *-commutative 40.1%

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

   8. Applied egg-rr 40.1%

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

      1. unpow2 40.1%

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

      2. rem-sqrt-square 53.7%

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

      3. *-commutative 53.7%

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

      4. associate-*l/ 50.5%

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

      5. associate-*r/ 53.8%

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

   10. Simplified 53.8%

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

   if -2e-281 < (sin.f64 ky) < 4.99999999999999969e-99

   1. Initial program 89.8%

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

      1. +-commutative 89.8%

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

      2. unpow2 89.8%

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

      3. unpow2 89.8%

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

      4. hypot-def 99.7%

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

   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 64.5%

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

   if 4.99999999999999969e-99 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-*l/ 91.2%

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

   2. associate-*r/ 93.3%

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

   3. +-commutative 93.3%

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

   4. unpow2 93.3%

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

   5. unpow2 93.3%

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

   6. hypot-def 99.6%

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

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 9: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;th \leq -2.55 \lor \neg \left(th \leq 9.5\right):\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (or (<= th -2.55) (not (<= th 9.5)))
     (/ (* ky (sin th)) t_1)
     (* (/ (sin ky) t_1) th))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if ((th <= -2.55) || !(th <= 9.5)) {
		tmp = (ky * sin(th)) / t_1;
	} else {
		tmp = (sin(ky) / t_1) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if ((th <= -2.55) || !(th <= 9.5)) {
		tmp = (ky * Math.sin(th)) / t_1;
	} else {
		tmp = (Math.sin(ky) / t_1) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if (th <= -2.55) or not (th <= 9.5):
		tmp = (ky * math.sin(th)) / t_1
	else:
		tmp = (math.sin(ky) / t_1) * th
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if ((th <= -2.55) || !(th <= 9.5))
		tmp = Float64(Float64(ky * sin(th)) / t_1);
	else
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if ((th <= -2.55) || ~((th <= 9.5)))
		tmp = (ky * sin(th)) / t_1;
	else
		tmp = (sin(ky) / t_1) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[th, -2.55], N[Not[LessEqual[th, 9.5]], $MachinePrecision]], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < -2.5499999999999998 or 9.5 < th

   1. Initial program 92.2%

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

      1. associate-*l/ 92.3%

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

      2. +-commutative 92.3%

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

      3. unpow2 92.3%

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

      4. unpow2 92.3%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 53.1%

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

   if -2.5499999999999998 < th < 9.5

   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. Taylor expanded in th around 0 98.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -2.55 \lor \neg \left(th \leq 9.5\right):\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \end{array} \]

Alternative 10: 47.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-71}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-71)
   (fabs (sin th))
   (if (<= (sin ky) 5e-99) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-71) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-99) {
		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-71)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-99) 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-71) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-99) {
		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-71:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-99:
		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-71)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-99)
		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-71)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-99)
		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-71], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-99], 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) < -9.9999999999999992e-72

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.2%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 25.2%

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

      3. pow2 25.2%

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

   6. Applied egg-rr 25.2%

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

      1. unpow2 25.2%

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

      2. rem-sqrt-square 34.7%

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

   8. Simplified 34.7%

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

   if -9.9999999999999992e-72 < (sin.f64 ky) < 4.99999999999999969e-99

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 52.2%

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

   if 4.99999999999999969e-99 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-71}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 47.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-71}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-71)
   (fabs (sin th))
   (if (<= (sin ky) 5e-99) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-71) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-99) {
		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-71)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-99) 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-71) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-99) {
		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-71:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-99:
		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-71)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-99)
		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-71)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-99)
		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-71], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-99], 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) < -9.9999999999999992e-72

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.2%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 25.2%

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

      3. pow2 25.2%

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

   6. Applied egg-rr 25.2%

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

      1. unpow2 25.2%

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

      2. rem-sqrt-square 34.7%

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

   8. Simplified 34.7%

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

   if -9.9999999999999992e-72 < (sin.f64 ky) < 4.99999999999999969e-99

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 52.5%

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

      1. associate-/l* 52.3%

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

   6. Simplified 52.3%

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

   if 4.99999999999999969e-99 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-71}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 46.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-57}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-57)
   (fabs (sin th))
   (if (<= (sin ky) 5e-99) (/ (* ky (sin th)) (sin kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-57) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-99) {
		tmp = (ky * sin(th)) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-57)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-99) then
        tmp = (ky * sin(th)) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-57) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-99) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-57:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-99:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-57)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-99)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-57)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-99)
		tmp = (ky * sin(th)) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-57], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-99], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.99999999999999955e-58

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. unpow2 99.8%

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

      4. hypot-def 99.8%

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

   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 3.1%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 25.6%

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

      3. pow2 25.6%

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

   6. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 34.2%

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

   8. Simplified 34.2%

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

   if -9.99999999999999955e-58 < (sin.f64 ky) < 4.99999999999999969e-99

   1. Initial program 83.1%

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

      1. +-commutative 83.1%

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

      2. unpow2 83.1%

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

      3. unpow2 83.1%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 52.5%

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

   if 4.99999999999999969e-99 < (sin.f64 ky)

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-57}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 33.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -1.6 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -6.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;ky \leq -1 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 940000000000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -1.6e-103)
     t_1
     (if (<= ky -6.9e-152)
       (/ ky (* (sin kx) (+ (/ 1.0 th) (* th 0.16666666666666666))))
       (if (<= ky -1e-228)
         t_1
         (if (<= ky 5e-97)
           (* th (fabs (/ ky (sin kx))))
           (if (<= ky 940000000000.0) (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -1.6e-103) {
		tmp = t_1;
	} else if (ky <= -6.9e-152) {
		tmp = ky / (sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (ky <= -1e-228) {
		tmp = t_1;
	} else if (ky <= 5e-97) {
		tmp = th * fabs((ky / sin(kx)));
	} else if (ky <= 940000000000.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (ky <= (-1.6d-103)) then
        tmp = t_1
    else if (ky <= (-6.9d-152)) then
        tmp = ky / (sin(kx) * ((1.0d0 / th) + (th * 0.16666666666666666d0)))
    else if (ky <= (-1d-228)) then
        tmp = t_1
    else if (ky <= 5d-97) then
        tmp = th * abs((ky / sin(kx)))
    else if (ky <= 940000000000.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -1.6e-103) {
		tmp = t_1;
	} else if (ky <= -6.9e-152) {
		tmp = ky / (Math.sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (ky <= -1e-228) {
		tmp = t_1;
	} else if (ky <= 5e-97) {
		tmp = th * Math.abs((ky / Math.sin(kx)));
	} else if (ky <= 940000000000.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -1.6e-103:
		tmp = t_1
	elif ky <= -6.9e-152:
		tmp = ky / (math.sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)))
	elif ky <= -1e-228:
		tmp = t_1
	elif ky <= 5e-97:
		tmp = th * math.fabs((ky / math.sin(kx)))
	elif ky <= 940000000000.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -1.6e-103)
		tmp = t_1;
	elseif (ky <= -6.9e-152)
		tmp = Float64(ky / Float64(sin(kx) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	elseif (ky <= -1e-228)
		tmp = t_1;
	elseif (ky <= 5e-97)
		tmp = Float64(th * abs(Float64(ky / sin(kx))));
	elseif (ky <= 940000000000.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -1.6e-103)
		tmp = t_1;
	elseif (ky <= -6.9e-152)
		tmp = ky / (sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)));
	elseif (ky <= -1e-228)
		tmp = t_1;
	elseif (ky <= 5e-97)
		tmp = th * abs((ky / sin(kx)));
	elseif (ky <= 940000000000.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -1.6e-103], t$95$1, If[LessEqual[ky, -6.9e-152], N[(ky / N[(N[Sin[kx], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, -1e-228], t$95$1, If[LessEqual[ky, 5e-97], N[(th * N[Abs[N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 940000000000.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -1.59999999999999988e-103 or -6.90000000000000039e-152 < ky < -1.00000000000000003e-228 or 9.4e11 < 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 99.7%

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

   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 20.4%

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

      1. add-sqr-sqrt 10.6%

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

      2. sqrt-unprod 26.7%

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

      3. pow2 26.7%

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

   6. Applied egg-rr 26.7%

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

      1. unpow2 26.7%

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

      2. rem-sqrt-square 32.2%

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

   8. Simplified 32.2%

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

   if -1.59999999999999988e-103 < ky < -6.90000000000000039e-152

   1. Initial program 100.0%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. unpow2 100.0%

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

      5. hypot-def 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in th around 0 57.0%

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

      1. *-commutative 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in ky around 0 57.7%

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

   if -1.00000000000000003e-228 < ky < 4.9999999999999995e-97

   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 99.7%

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

   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 th around 0 57.6%

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

   5. Taylor expanded in ky around 0 37.5%

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

      1. add-sqr-sqrt 20.0%

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

      2. sqrt-unprod 41.3%

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

      3. pow2 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. unpow2 41.3%

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

      2. rem-sqrt-square 45.9%

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

   9. Simplified 45.9%

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

   if 4.9999999999999995e-97 < ky < 9.4e11

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in kx around 0 50.5%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.6 \cdot 10^{-103}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -6.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;ky \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 940000000000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]

Alternative 14: 33.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -4.5 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq -1.05 \cdot 10^{-150}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;ky \leq -3 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{elif}\;ky \leq 940000000000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))))
   (if (<= ky -4.5e-104)
     t_1
     (if (<= ky -1.05e-150)
       (/ ky (* (sin kx) (+ (/ 1.0 th) (* th 0.16666666666666666))))
       (if (<= ky -3e-205)
         t_1
         (if (<= ky 3.6e-166)
           (/ (* ky (sin th)) kx)
           (if (<= ky 940000000000.0) (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double tmp;
	if (ky <= -4.5e-104) {
		tmp = t_1;
	} else if (ky <= -1.05e-150) {
		tmp = ky / (sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (ky <= -3e-205) {
		tmp = t_1;
	} else if (ky <= 3.6e-166) {
		tmp = (ky * sin(th)) / kx;
	} else if (ky <= 940000000000.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(sin(th))
    if (ky <= (-4.5d-104)) then
        tmp = t_1
    else if (ky <= (-1.05d-150)) then
        tmp = ky / (sin(kx) * ((1.0d0 / th) + (th * 0.16666666666666666d0)))
    else if (ky <= (-3d-205)) then
        tmp = t_1
    else if (ky <= 3.6d-166) then
        tmp = (ky * sin(th)) / kx
    else if (ky <= 940000000000.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double tmp;
	if (ky <= -4.5e-104) {
		tmp = t_1;
	} else if (ky <= -1.05e-150) {
		tmp = ky / (Math.sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (ky <= -3e-205) {
		tmp = t_1;
	} else if (ky <= 3.6e-166) {
		tmp = (ky * Math.sin(th)) / kx;
	} else if (ky <= 940000000000.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	tmp = 0
	if ky <= -4.5e-104:
		tmp = t_1
	elif ky <= -1.05e-150:
		tmp = ky / (math.sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)))
	elif ky <= -3e-205:
		tmp = t_1
	elif ky <= 3.6e-166:
		tmp = (ky * math.sin(th)) / kx
	elif ky <= 940000000000.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	tmp = 0.0
	if (ky <= -4.5e-104)
		tmp = t_1;
	elseif (ky <= -1.05e-150)
		tmp = Float64(ky / Float64(sin(kx) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	elseif (ky <= -3e-205)
		tmp = t_1;
	elseif (ky <= 3.6e-166)
		tmp = Float64(Float64(ky * sin(th)) / kx);
	elseif (ky <= 940000000000.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	tmp = 0.0;
	if (ky <= -4.5e-104)
		tmp = t_1;
	elseif (ky <= -1.05e-150)
		tmp = ky / (sin(kx) * ((1.0 / th) + (th * 0.16666666666666666)));
	elseif (ky <= -3e-205)
		tmp = t_1;
	elseif (ky <= 3.6e-166)
		tmp = (ky * sin(th)) / kx;
	elseif (ky <= 940000000000.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, -4.5e-104], t$95$1, If[LessEqual[ky, -1.05e-150], N[(ky / N[(N[Sin[kx], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, -3e-205], t$95$1, If[LessEqual[ky, 3.6e-166], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], If[LessEqual[ky, 940000000000.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < -4.4999999999999997e-104 or -1.0500000000000001e-150 < ky < -3e-205 or 9.4e11 < ky

   1. Initial program 95.7%

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

      1. +-commutative 95.7%

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

      2. unpow2 95.7%

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

      3. unpow2 95.7%

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

      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 21.1%

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

      1. add-sqr-sqrt 11.1%

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

      2. sqrt-unprod 27.1%

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

      3. pow2 27.1%

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

   6. Applied egg-rr 27.1%

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

      1. unpow2 27.1%

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

      2. rem-sqrt-square 32.8%

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

   8. Simplified 32.8%

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

   if -4.4999999999999997e-104 < ky < -1.0500000000000001e-150

   1. Initial program 100.0%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. unpow2 100.0%

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

      5. hypot-def 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in th around 0 57.0%

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

      1. *-commutative 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in ky around 0 57.7%

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

   if -3e-205 < ky < 3.6000000000000001e-166

   1. Initial program 82.3%

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

      1. +-commutative 82.3%

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

      2. unpow2 82.3%

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

      3. unpow2 82.3%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 61.9%

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

      1. associate-/l* 61.7%

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

   6. Simplified 61.7%

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

   7. Taylor expanded in kx around 0 46.3%

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

   if 3.6000000000000001e-166 < ky < 9.4e11

   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 99.6%

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

   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 43.2%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -4.5 \cdot 10^{-104}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -1.05 \cdot 10^{-150}:\\ \;\;\;\;\frac{ky}{\sin kx \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;ky \leq -3 \cdot 10^{-205}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{elif}\;ky \leq 940000000000:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]

Alternative 15: 31.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;\left|th \cdot \frac{ky}{kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -10500000000.0)
   (sin th)
   (if (<= ky 4.2e-174) (fabs (* th (/ ky kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = sin(th);
	} else if (ky <= 4.2e-174) {
		tmp = fabs((th * (ky / kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-10500000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 4.2d-174) then
        tmp = abs((th * (ky / kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 4.2e-174) {
		tmp = Math.abs((th * (ky / kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -10500000000.0:
		tmp = math.sin(th)
	elif ky <= 4.2e-174:
		tmp = math.fabs((th * (ky / kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 4.2e-174)
		tmp = abs(Float64(th * Float64(ky / kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 4.2e-174)
		tmp = abs((th * (ky / kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -10500000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 4.2e-174], N[Abs[N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.05e10 or 4.20000000000000021e-174 < ky

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      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. Taylor expanded in kx around 0 29.3%

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

   if -1.05e10 < ky < 4.20000000000000021e-174

   1. Initial program 82.4%

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

      1. +-commutative 82.4%

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

      2. unpow2 82.4%

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

      3. unpow2 82.4%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 48.9%

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

   5. Taylor expanded in ky around 0 29.6%

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

   6. Taylor expanded in kx around 0 27.9%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 27.2%

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

      3. pow2 27.2%

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

      4. *-commutative 27.2%

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

   8. Applied egg-rr 27.2%

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

      1. unpow2 27.2%

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

      2. rem-sqrt-square 29.5%

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

   10. Simplified 29.5%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;\left|th \cdot \frac{ky}{kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 32.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -6200000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.9 \cdot 10^{-173}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -6200000000.0)
   (sin th)
   (if (<= ky 1.9e-173) (* th (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -6200000000.0) {
		tmp = sin(th);
	} else if (ky <= 1.9e-173) {
		tmp = 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 (ky <= (-6200000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 1.9d-173) then
        tmp = 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 (ky <= -6200000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1.9e-173) {
		tmp = th * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -6200000000.0:
		tmp = math.sin(th)
	elif ky <= 1.9e-173:
		tmp = th * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -6200000000.0)
		tmp = sin(th);
	elseif (ky <= 1.9e-173)
		tmp = Float64(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 (ky <= -6200000000.0)
		tmp = sin(th);
	elseif (ky <= 1.9e-173)
		tmp = th * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -6200000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.9e-173], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -6.2e9 or 1.90000000000000015e-173 < ky

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      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. Taylor expanded in kx around 0 29.5%

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

   if -6.2e9 < ky < 1.90000000000000015e-173

   1. Initial program 82.6%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. unpow2 82.6%

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

      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 th around 0 49.5%

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

   5. Taylor expanded in ky around 0 30.4%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6200000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.9 \cdot 10^{-173}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 33.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -10500000000.0)
   (sin th)
   (if (<= ky 3.2e-166) (/ (sin th) (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = sin(th);
	} else if (ky <= 3.2e-166) {
		tmp = sin(th) / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-10500000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 3.2d-166) then
        tmp = sin(th) / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 3.2e-166) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -10500000000.0:
		tmp = math.sin(th)
	elif ky <= 3.2e-166:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 3.2e-166)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 3.2e-166)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -10500000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.2e-166], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.05e10 or 3.20000000000000001e-166 < ky

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      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. Taylor expanded in kx around 0 29.6%

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

   if -1.05e10 < ky < 3.20000000000000001e-166

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. hypot-def 99.7%

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

   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 48.9%

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

      1. associate-/l* 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in kx around 0 33.8%

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

      1. associate-/l* 33.6%

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

   9. Simplified 33.6%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18: 33.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -10500000000.0)
   (sin th)
   (if (<= ky 3.8e-166) (/ (* ky (sin th)) kx) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = sin(th);
	} else if (ky <= 3.8e-166) {
		tmp = (ky * sin(th)) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-10500000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 3.8d-166) then
        tmp = (ky * sin(th)) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 3.8e-166) {
		tmp = (ky * Math.sin(th)) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -10500000000.0:
		tmp = math.sin(th)
	elif ky <= 3.8e-166:
		tmp = (ky * math.sin(th)) / kx
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 3.8e-166)
		tmp = Float64(Float64(ky * sin(th)) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 3.8e-166)
		tmp = (ky * sin(th)) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -10500000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.8e-166], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.05e10 or 3.79999999999999982e-166 < ky

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      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. Taylor expanded in kx around 0 29.6%

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

   if -1.05e10 < ky < 3.79999999999999982e-166

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. hypot-def 99.7%

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

   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 48.9%

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

      1. associate-/l* 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in kx around 0 33.8%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19: 30.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -10500000000.0)
   (sin th)
   (if (<= ky 2.8e-166) (/ (* ky th) kx) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = sin(th);
	} else if (ky <= 2.8e-166) {
		tmp = (ky * th) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-10500000000.0d0)) then
        tmp = sin(th)
    else if (ky <= 2.8d-166) then
        tmp = (ky * th) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -10500000000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 2.8e-166) {
		tmp = (ky * th) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -10500000000.0:
		tmp = math.sin(th)
	elif ky <= 2.8e-166:
		tmp = (ky * th) / kx
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 2.8e-166)
		tmp = Float64(Float64(ky * th) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -10500000000.0)
		tmp = sin(th);
	elseif (ky <= 2.8e-166)
		tmp = (ky * th) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -10500000000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 2.8e-166], N[(N[(ky * th), $MachinePrecision] / kx), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.05e10 or 2.7999999999999999e-166 < ky

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      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. Taylor expanded in kx around 0 29.6%

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

   if -1.05e10 < ky < 2.7999999999999999e-166

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. hypot-def 99.7%

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

   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 th around 0 48.9%

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

   5. Taylor expanded in ky around 0 30.1%

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

   6. Taylor expanded in kx around 0 28.6%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -10500000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 20: 21.6% accurate, 54.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.3 \cdot 10^{-8} \lor \neg \left(ky \leq 1.8 \cdot 10^{-160}\right):\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -1.3e-8) (not (<= ky 1.8e-160)))
   (/ 1.0 (+ (/ 1.0 th) (* th 0.16666666666666666)))
   (/ (* ky th) kx)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.3e-8) || !(ky <= 1.8e-160)) {
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
	} else {
		tmp = (ky * th) / kx;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-1.3d-8)) .or. (.not. (ky <= 1.8d-160))) then
        tmp = 1.0d0 / ((1.0d0 / th) + (th * 0.16666666666666666d0))
    else
        tmp = (ky * th) / kx
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1.3e-8) || !(ky <= 1.8e-160)) {
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
	} else {
		tmp = (ky * th) / kx;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -1.3e-8) or not (ky <= 1.8e-160):
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666))
	else:
		tmp = (ky * th) / kx
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -1.3e-8) || !(ky <= 1.8e-160))
		tmp = Float64(1.0 / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)));
	else
		tmp = Float64(Float64(ky * th) / kx);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -1.3e-8) || ~((ky <= 1.8e-160)))
		tmp = 1.0 / ((1.0 / th) + (th * 0.16666666666666666));
	else
		tmp = (ky * th) / kx;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -1.3e-8], N[Not[LessEqual[ky, 1.8e-160]], $MachinePrecision]], N[(1.0 / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / kx), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.3000000000000001e-8 or 1.7999999999999999e-160 < ky

   1. Initial program 99.3%

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

      1. associate-/r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. unpow2 99.2%

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

      4. unpow2 99.2%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 55.3%

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

      1. *-commutative 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in kx around 0 18.5%

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

   if -1.3000000000000001e-8 < ky < 1.7999999999999999e-160

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. unpow2 81.9%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 49.6%

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

   5. Taylor expanded in ky around 0 30.9%

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

   6. Taylor expanded in kx around 0 29.4%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.3 \cdot 10^{-8} \lor \neg \left(ky \leq 1.8 \cdot 10^{-160}\right):\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \end{array} \]

Alternative 21: 22.0% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.7 \cdot 10^{-160}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.1e-6) th (if (<= ky 1.7e-160) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.1e-6) {
		tmp = th;
	} else if (ky <= 1.7e-160) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.1d-6)) then
        tmp = th
    else if (ky <= 1.7d-160) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.1e-6) {
		tmp = th;
	} else if (ky <= 1.7e-160) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.1e-6:
		tmp = th
	elif ky <= 1.7e-160:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.1e-6)
		tmp = th;
	elseif (ky <= 1.7e-160)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.1e-6)
		tmp = th;
	elseif (ky <= 1.7e-160)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.1e-6], th, If[LessEqual[ky, 1.7e-160], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.1000000000000001e-6 or 1.70000000000000011e-160 < ky

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. hypot-def 99.6%

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

   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 th around 0 55.1%

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

   5. Taylor expanded in kx around 0 18.2%

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

   if -1.1000000000000001e-6 < ky < 1.70000000000000011e-160

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. unpow2 81.9%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 49.6%

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

   5. Taylor expanded in ky around 0 30.9%

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

   6. Taylor expanded in kx around 0 29.4%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.7 \cdot 10^{-160}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 22: 21.3% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -5.5e-7) th (if (<= ky 1.8e-160) (/ (* ky th) kx) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.5e-7) {
		tmp = th;
	} else if (ky <= 1.8e-160) {
		tmp = (ky * th) / kx;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-5.5d-7)) then
        tmp = th
    else if (ky <= 1.8d-160) then
        tmp = (ky * th) / kx
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -5.5e-7) {
		tmp = th;
	} else if (ky <= 1.8e-160) {
		tmp = (ky * th) / kx;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -5.5e-7:
		tmp = th
	elif ky <= 1.8e-160:
		tmp = (ky * th) / kx
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -5.5e-7)
		tmp = th;
	elseif (ky <= 1.8e-160)
		tmp = Float64(Float64(ky * th) / kx);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -5.5e-7)
		tmp = th;
	elseif (ky <= 1.8e-160)
		tmp = (ky * th) / kx;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -5.5e-7], th, If[LessEqual[ky, 1.8e-160], N[(N[(ky * th), $MachinePrecision] / kx), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -5.5000000000000003e-7 or 1.7999999999999999e-160 < ky

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. hypot-def 99.6%

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

   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 th around 0 55.1%

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

   5. Taylor expanded in kx around 0 18.2%

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

   if -5.5000000000000003e-7 < ky < 1.7999999999999999e-160

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. unpow2 81.9%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 49.6%

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

   5. Taylor expanded in ky around 0 30.9%

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

   6. Taylor expanded in kx around 0 29.4%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{ky \cdot th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 23: 13.7% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.4%

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

   1. +-commutative 93.4%

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

   2. unpow2 93.4%

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

   3. unpow2 93.4%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in th around 0 53.3%

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

5. Taylor expanded in kx around 0 13.4%

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

6. Final simplification 13.4%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023178 
(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)))