Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 17.8s

Alternatives: 18

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. clear-num 93.9%

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

   2. associate-*l/ 94.0%

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

   3. *-un-lft-identity 94.0%

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

   4. unpow2 94.0%

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

   5. unpow2 94.0%

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

   6. hypot-def 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 46.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-80}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (fabs (sin th))
   (if (<= (sin ky) 1e-80) (fabs (* ky (/ (sin th) (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-80) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-80) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-80) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-80:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-80)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-80)
		tmp = abs((ky * (sin(th) / sin(kx))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-80], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. clear-num 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. hypot-def 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0 2.7%

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

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 29.3%

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

      4. pow2 29.3%

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

   7. Applied egg-rr 29.3%

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

      1. unpow2 29.3%

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

      2. rem-sqrt-square 37.5%

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

   9. Simplified 37.5%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.99999999999999961e-81

   1. Initial program 87.3%

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

   3. Taylor expanded in ky around 0 46.1%

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

      1. add-sqr-sqrt 31.6%

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

      2. sqrt-unprod 35.7%

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

      3. pow2 35.7%

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

      4. associate-/l* 36.4%

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

      5. associate-/r/ 36.5%

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

   5. Applied egg-rr 36.5%

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

      1. unpow2 36.5%

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

      2. rem-sqrt-square 49.0%

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

      3. associate-*l/ 47.6%

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

      4. associate-*r/ 49.0%

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

   7. Simplified 49.0%

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

   if 9.99999999999999961e-81 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0 66.2%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-80}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (fabs (sin th))
   (if (<= (sin ky) 2e-50) (/ (* (sin th) ky) (fabs (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-50) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-50) then
        tmp = (sin(th) * ky) / abs(sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-50) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-50:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-50)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-50)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-50], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. clear-num 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. hypot-def 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0 2.7%

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

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 29.3%

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

      4. pow2 29.3%

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

   7. Applied egg-rr 29.3%

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

      1. unpow2 29.3%

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

      2. rem-sqrt-square 37.5%

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

   9. Simplified 37.5%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 2.00000000000000002e-50

   1. Initial program 88.2%

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

   3. Taylor expanded in ky around 0 46.5%

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

      1. add-sqr-sqrt 37.2%

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

      2. sqrt-prod 70.8%

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

      3. rem-sqrt-square 75.4%

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

   5. Applied egg-rr 75.4%

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

   if 2.00000000000000002e-50 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0 68.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;kx \leq 3.8 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{e^{\log \left(\left|\sin kx\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 130.0)
   (/ (sin th) (/ (hypot kx (sin ky)) (sin ky)))
   (if (<= kx 3.8e+248)
     (/ (* (sin th) ky) (exp (log (fabs (sin kx)))))
     (/
      (sin ky)
      (*
       (hypot (sin kx) (sin ky))
       (+ (/ 1.0 th) (* th 0.16666666666666666)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = sin(th) / (hypot(kx, sin(ky)) / sin(ky));
	} else if (kx <= 3.8e+248) {
		tmp = (sin(th) * ky) / exp(log(fabs(sin(kx))));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = Math.sin(th) / (Math.hypot(kx, Math.sin(ky)) / Math.sin(ky));
	} else if (kx <= 3.8e+248) {
		tmp = (Math.sin(th) * ky) / Math.exp(Math.log(Math.abs(Math.sin(kx))));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 130.0:
		tmp = math.sin(th) / (math.hypot(kx, math.sin(ky)) / math.sin(ky))
	elif kx <= 3.8e+248:
		tmp = (math.sin(th) * ky) / math.exp(math.log(math.fabs(math.sin(kx))))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 130.0)
		tmp = Float64(sin(th) / Float64(hypot(kx, sin(ky)) / sin(ky)));
	elseif (kx <= 3.8e+248)
		tmp = Float64(Float64(sin(th) * ky) / exp(log(abs(sin(kx)))));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 130.0)
		tmp = sin(th) / (hypot(kx, sin(ky)) / sin(ky));
	elseif (kx <= 3.8e+248)
		tmp = (sin(th) * ky) / exp(log(abs(sin(kx))));
	else
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 130.0], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 3.8e+248], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Exp[N[Log[N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 130

   1. Initial program 92.4%

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

      1. clear-num 92.4%

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

      2. associate-*l/ 92.4%

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

      3. *-un-lft-identity 92.4%

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

      4. unpow2 92.4%

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

      5. unpow2 92.4%

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

      6. hypot-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0 73.1%

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

   if 130 < kx < 3.8000000000000001e248

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 31.3%

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

      1. add-exp-log 29.2%

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

   5. Applied egg-rr 29.2%

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

      1. add-sqr-sqrt 29.2%

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

      2. sqrt-prod 56.3%

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

      3. rem-sqrt-square 56.3%

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

   7. Applied egg-rr 56.3%

      \[\leadsto \frac{ky \cdot \sin th}{e^{\log \color{blue}{\left(\left|\sin kx\right|\right)}}} \]

   if 3.8000000000000001e248 < kx

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. *-commutative 63.0%

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

      3. unpow2 63.0%

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

      4. unpow2 63.0%

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

      5. hypot-def 63.0%

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

      6. associate-*r/ 63.0%

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

      7. *-rgt-identity 63.0%

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

      8. *-lft-identity 63.0%

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

      9. associate-*l/ 63.0%

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

      10. associate-*r* 63.0%

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

   7. Simplified 63.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;kx \leq 3.8 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{e^{\log \left(\left|\sin kx\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. expm1-log1p-u 93.8%

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

   2. expm1-udef 42.1%

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

4. Applied egg-rr 44.1%

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

   1. expm1-def 99.6%

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

   2. expm1-log1p 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r/ 95.8%

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

   5. associate-*l/ 99.6%

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

   6. *-commutative 99.6%

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

   7. hypot-def 93.8%

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

   8. unpow2 93.8%

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

   9. unpow2 93.8%

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

   10. +-commutative 93.8%

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

   11. unpow2 93.8%

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

   12. unpow2 93.8%

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

   13. hypot-def 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. remove-double-neg 93.9%

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

   2. sin-neg 93.9%

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

   3. neg-mul-1 93.9%

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

   4. *-commutative 93.9%

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

   5. associate-*l* 93.9%

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

   6. associate-*l/ 91.9%

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

   7. associate-/r/ 91.9%

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

   8. associate-*l/ 93.9%

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

   9. associate-/r/ 93.9%

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

   10. sin-neg 93.9%

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

   11. neg-mul-1 93.9%

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

   12. associate-/r* 93.9%

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

   13. associate-/r/ 93.9%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;kx \leq 3 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 130.0)
   (/ (sin th) (/ (hypot kx (sin ky)) (sin ky)))
   (if (<= kx 3e+248)
     (/ (* (sin th) ky) (fabs (sin kx)))
     (/
      (sin ky)
      (*
       (hypot (sin kx) (sin ky))
       (+ (/ 1.0 th) (* th 0.16666666666666666)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = sin(th) / (hypot(kx, sin(ky)) / sin(ky));
	} else if (kx <= 3e+248) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = Math.sin(th) / (Math.hypot(kx, Math.sin(ky)) / Math.sin(ky));
	} else if (kx <= 3e+248) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 130.0:
		tmp = math.sin(th) / (math.hypot(kx, math.sin(ky)) / math.sin(ky))
	elif kx <= 3e+248:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 130.0)
		tmp = Float64(sin(th) / Float64(hypot(kx, sin(ky)) / sin(ky)));
	elseif (kx <= 3e+248)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 130.0)
		tmp = sin(th) / (hypot(kx, sin(ky)) / sin(ky));
	elseif (kx <= 3e+248)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 130.0], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 3e+248], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 130

   1. Initial program 92.4%

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

      1. clear-num 92.4%

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

      2. associate-*l/ 92.4%

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

      3. *-un-lft-identity 92.4%

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

      4. unpow2 92.4%

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

      5. unpow2 92.4%

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

      6. hypot-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0 73.1%

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

   if 130 < kx < 3e248

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 31.3%

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

      1. add-sqr-sqrt 29.2%

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

      2. sqrt-prod 56.3%

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

      3. rem-sqrt-square 56.3%

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

   5. Applied egg-rr 56.3%

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

   if 3e248 < kx

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. *-commutative 63.0%

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

      3. unpow2 63.0%

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

      4. unpow2 63.0%

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

      5. hypot-def 63.0%

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

      6. associate-*r/ 63.0%

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

      7. *-rgt-identity 63.0%

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

      8. *-lft-identity 63.0%

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

      9. associate-*l/ 63.0%

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

      10. associate-*r* 63.0%

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

   7. Simplified 63.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + 0.16666666666666666 \cdot th\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;kx \leq 3 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 130.0)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (if (<= kx 3.1e+248)
     (/ (* (sin th) ky) (fabs (sin kx)))
     (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (kx <= 3.1e+248) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (kx <= 3.1e+248) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 130.0:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif kx <= 3.1e+248:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 130.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (kx <= 3.1e+248)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 130.0)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (kx <= 3.1e+248)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 130.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 3.1e+248], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 130

   1. Initial program 92.4%

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

      1. remove-double-neg 92.4%

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

      2. sin-neg 92.4%

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

      3. neg-mul-1 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*l* 92.4%

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

      6. associate-*l/ 89.8%

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

      7. associate-/r/ 89.8%

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

      8. associate-*l/ 92.4%

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

      9. associate-/r/ 92.3%

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

      10. sin-neg 92.3%

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

      11. neg-mul-1 92.3%

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

      12. associate-/r* 92.3%

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

      13. associate-/r/ 92.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 73.0%

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

   if 130 < kx < 3.10000000000000005e248

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 31.3%

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

      1. add-sqr-sqrt 29.2%

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

      2. sqrt-prod 56.3%

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

      3. rem-sqrt-square 56.3%

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

   5. Applied egg-rr 56.3%

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

   if 3.10000000000000005e248 < kx

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0 62.6%

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

      1. associate-*l/ 62.6%

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

      2. unpow2 62.6%

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

      3. unpow2 62.6%

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

      4. hypot-def 62.6%

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

      5. *-lft-identity 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;kx \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 130.0)
   (/ (sin th) (/ (hypot kx (sin ky)) (sin ky)))
   (if (<= kx 3.1e+248)
     (/ (* (sin th) ky) (fabs (sin kx)))
     (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = sin(th) / (hypot(kx, sin(ky)) / sin(ky));
	} else if (kx <= 3.1e+248) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = Math.sin(th) / (Math.hypot(kx, Math.sin(ky)) / Math.sin(ky));
	} else if (kx <= 3.1e+248) {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 130.0:
		tmp = math.sin(th) / (math.hypot(kx, math.sin(ky)) / math.sin(ky))
	elif kx <= 3.1e+248:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 130.0)
		tmp = Float64(sin(th) / Float64(hypot(kx, sin(ky)) / sin(ky)));
	elseif (kx <= 3.1e+248)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 130.0)
		tmp = sin(th) / (hypot(kx, sin(ky)) / sin(ky));
	elseif (kx <= 3.1e+248)
		tmp = (sin(th) * ky) / abs(sin(kx));
	else
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 130.0], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 3.1e+248], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 130

   1. Initial program 92.4%

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

      1. clear-num 92.4%

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

      2. associate-*l/ 92.4%

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

      3. *-un-lft-identity 92.4%

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

      4. unpow2 92.4%

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

      5. unpow2 92.4%

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

      6. hypot-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0 73.1%

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

   if 130 < kx < 3.10000000000000005e248

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 31.3%

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

      1. add-sqr-sqrt 29.2%

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

      2. sqrt-prod 56.3%

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

      3. rem-sqrt-square 56.3%

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

   5. Applied egg-rr 56.3%

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

   if 3.10000000000000005e248 < kx

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.6%

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

      3. unpow2 99.6%

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

      4. unpow2 99.6%

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

      5. hypot-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0 62.6%

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

      1. associate-*l/ 62.6%

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

      2. unpow2 62.6%

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

      3. unpow2 62.6%

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

      4. hypot-def 62.6%

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

      5. *-lft-identity 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;kx \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (fabs (sin th))
   (if (<= (sin ky) 5e-100) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-100) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-100) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-100) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-100:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-100)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-100)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-100], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. associate-*l/ 99.4%

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

      2. clear-num 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. hypot-def 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0 2.7%

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

      1. remove-double-div 2.7%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 29.3%

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

      4. pow2 29.3%

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

   7. Applied egg-rr 29.3%

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

      1. unpow2 29.3%

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

      2. rem-sqrt-square 37.5%

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

   9. Simplified 37.5%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 5.0000000000000001e-100

   1. Initial program 87.1%

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

      1. clear-num 87.1%

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

      2. associate-*l/ 87.2%

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

      3. *-un-lft-identity 87.2%

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

      4. unpow2 87.2%

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

      5. unpow2 87.2%

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

      6. hypot-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ky around 0 49.7%

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

   if 5.0000000000000001e-100 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0 66.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 130.0)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (/ (* (sin th) ky) (fabs (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 130.0) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = (Math.sin(th) * ky) / Math.abs(Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 130.0:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = (math.sin(th) * ky) / math.fabs(math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 130.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 130.0)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = (sin(th) * ky) / abs(sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 130.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 130

   1. Initial program 92.4%

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

      1. remove-double-neg 92.4%

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

      2. sin-neg 92.4%

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

      3. neg-mul-1 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*l* 92.4%

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

      6. associate-*l/ 89.8%

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

      7. associate-/r/ 89.8%

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

      8. associate-*l/ 92.4%

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

      9. associate-/r/ 92.3%

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

      10. sin-neg 92.3%

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

      11. neg-mul-1 92.3%

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

      12. associate-/r* 92.3%

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

      13. associate-/r/ 92.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 73.0%

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

   if 130 < kx

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 35.1%

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

      1. add-sqr-sqrt 29.6%

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

      2. sqrt-prod 55.9%

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

      3. rem-sqrt-square 55.9%

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

   5. Applied egg-rr 55.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 130:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;ky \leq 9.2 \cdot 10^{+19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.5e-115)
   (/ ky (/ kx (sin th)))
   (if (<= ky 9.2e+19) (sin th) (fabs (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.5e-115) {
		tmp = ky / (kx / sin(th));
	} else if (ky <= 9.2e+19) {
		tmp = sin(th);
	} else {
		tmp = fabs(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 <= 4.5d-115) then
        tmp = ky / (kx / sin(th))
    else if (ky <= 9.2d+19) then
        tmp = sin(th)
    else
        tmp = abs(sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.5e-115) {
		tmp = ky / (kx / Math.sin(th));
	} else if (ky <= 9.2e+19) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.abs(Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.5e-115:
		tmp = ky / (kx / math.sin(th))
	elif ky <= 9.2e+19:
		tmp = math.sin(th)
	else:
		tmp = math.fabs(math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.5e-115)
		tmp = Float64(ky / Float64(kx / sin(th)));
	elseif (ky <= 9.2e+19)
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.5e-115)
		tmp = ky / (kx / sin(th));
	elseif (ky <= 9.2e+19)
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.5e-115], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 9.2e+19], N[Sin[th], $MachinePrecision], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.50000000000000023e-115

   1. Initial program 91.1%

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

   3. Taylor expanded in ky around 0 32.0%

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

   4. Taylor expanded in kx around 0 21.9%

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

      1. associate-/l* 24.1%

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

   6. Simplified 24.1%

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

   if 4.50000000000000023e-115 < ky < 9.2e19

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0 58.2%

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

   if 9.2e19 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. hypot-def 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0 30.4%

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

      1. remove-double-div 30.5%

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

      2. add-sqr-sqrt 11.6%

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

      3. sqrt-unprod 21.7%

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

      4. pow2 21.7%

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

   7. Applied egg-rr 21.7%

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

      1. unpow2 21.7%

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

      2. rem-sqrt-square 32.3%

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

   9. Simplified 32.3%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{elif}\;ky \leq 9.2 \cdot 10^{+19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 10^{+20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 7.8e-100)
   (* ky (/ (sin th) (sin kx)))
   (if (<= ky 1e+20) (sin th) (fabs (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7.8e-100) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (ky <= 1e+20) {
		tmp = sin(th);
	} else {
		tmp = fabs(sin(th));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 7.8d-100) then
        tmp = ky * (sin(th) / sin(kx))
    else if (ky <= 1d+20) then
        tmp = sin(th)
    else
        tmp = abs(sin(th))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7.8e-100) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else if (ky <= 1e+20) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.abs(Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 7.8e-100:
		tmp = ky * (math.sin(th) / math.sin(kx))
	elif ky <= 1e+20:
		tmp = math.sin(th)
	else:
		tmp = math.fabs(math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 7.8e-100)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (ky <= 1e+20)
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 7.8e-100)
		tmp = ky * (sin(th) / sin(kx));
	elseif (ky <= 1e+20)
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 7.8e-100], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 1e+20], N[Sin[th], $MachinePrecision], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 7.79999999999999955e-100

   1. Initial program 91.2%

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

      1. clear-num 91.2%

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

      2. associate-*l/ 91.3%

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

      3. *-un-lft-identity 91.3%

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

      4. unpow2 91.3%

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

      5. unpow2 91.3%

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

      6. hypot-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ky around 0 32.1%

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

      1. associate-*r/ 34.2%

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

   7. Simplified 34.2%

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

   if 7.79999999999999955e-100 < ky < 1e20

   1. Initial program 99.9%

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

   3. Taylor expanded in kx around 0 64.3%

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

   if 1e20 < ky

   1. Initial program 99.7%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. hypot-def 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in kx around 0 30.4%

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

      1. remove-double-div 30.5%

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

      2. add-sqr-sqrt 11.6%

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

      3. sqrt-unprod 21.7%

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

      4. pow2 21.7%

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

   7. Applied egg-rr 21.7%

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

      1. unpow2 21.7%

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

      2. rem-sqrt-square 32.3%

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

   9. Simplified 32.3%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 10^{+20}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.8 \cdot 10^{-100}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.8e-100) (* ky (/ th (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.8e-100) {
		tmp = ky * (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 (ky <= 3.8d-100) then
        tmp = ky * (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 (ky <= 3.8e-100) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.8e-100:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.8e-100)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3.8e-100)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.8e-100], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 3.79999999999999997e-100

   1. Initial program 91.2%

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

      1. associate-*l/ 88.8%

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

      2. clear-num 88.2%

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

      3. unpow2 88.2%

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

      4. unpow2 88.2%

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

      5. hypot-def 94.0%

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in ky around 0 31.6%

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

   6. Taylor expanded in th around 0 20.1%

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

      1. *-lft-identity 20.1%

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

      2. times-frac 22.2%

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

      3. /-rgt-identity 22.2%

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

   8. Simplified 22.2%

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

   if 3.79999999999999997e-100 < ky

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0 41.2%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.8 \cdot 10^{-100}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.2e-114) (/ ky (/ kx (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.2e-114) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 3.2d-114) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.2e-114) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3.2e-114:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.2e-114)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3.2e-114)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.2e-114], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 3.2000000000000002e-114

   1. Initial program 91.1%

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

   3. Taylor expanded in ky around 0 32.0%

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

   4. Taylor expanded in kx around 0 21.9%

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

      1. associate-/l* 24.1%

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

   6. Simplified 24.1%

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

   if 3.2000000000000002e-114 < ky

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0 39.9%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in kx around 0 26.0%

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

4. Final simplification 26.0%

   \[\leadsto \sin th \]
5. Add Preprocessing

Alternative 17: 14.2% accurate, 78.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. associate-*l/ 91.9%

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

   2. clear-num 91.4%

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

   3. unpow2 91.4%

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

   4. unpow2 91.4%

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

   5. hypot-def 95.4%

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

4. Applied egg-rr 95.4%

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

5. Taylor expanded in kx around 0 26.0%

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

6. Taylor expanded in th around 0 15.0%

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

7. Final simplification 15.0%

   \[\leadsto \frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666} \]
8. Add Preprocessing

Alternative 18: 13.5% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.9%

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

   1. associate-*l/ 91.9%

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

   2. clear-num 91.4%

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

   3. unpow2 91.4%

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

   4. unpow2 91.4%

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

   5. hypot-def 95.4%

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

4. Applied egg-rr 95.4%

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

5. Taylor expanded in kx around 0 26.0%

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

6. Taylor expanded in th around 0 14.2%

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

7. Final simplification 14.2%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))