Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 24.0s

Alternatives: 17

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $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. Step-by-step derivation

   1. +-commutative 93.9%

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

   2. unpow2 93.9%

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

   3. unpow2 93.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

   1. *-commutative 99.7%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 48.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.854:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.854)
   (/ (sin ky) (fabs (/ (sin ky) (sin th))))
   (if (<= (sin ky) -1e-30)
     (fabs (sin th))
     (if (<= (sin ky) 1e-114) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.854) {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	} else if (sin(ky) <= -1e-30) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-114) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.854d0)) then
        tmp = sin(ky) / abs((sin(ky) / sin(th)))
    else if (sin(ky) <= (-1d-30)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-114) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.854) {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	} else if (Math.sin(ky) <= -1e-30) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-114) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.854:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	elif math.sin(ky) <= -1e-30:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-114:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.854)
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	elseif (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.854)
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	elseif (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.854], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-30], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.853999999999999981

   1. Initial program 99.9%

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

      1. associate-/r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. sqr-neg 99.7%

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

      5. sin-neg 99.7%

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

      6. sin-neg 99.7%

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

      7. unpow2 99.7%

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

      8. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.0%

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

      1. add-sqr-sqrt 0.6%

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

      2. sqrt-unprod 63.0%

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

      3. pow2 63.0%

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

   6. Applied egg-rr 63.0%

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

      1. unpow2 63.0%

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

      2. rem-sqrt-square 67.7%

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

   8. Simplified 67.7%

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

   if -0.853999999999999981 < (sin.f64 ky) < -1e-30

   1. Initial program 99.5%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. sqr-neg 99.5%

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

      5. sin-neg 99.5%

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

      6. sin-neg 99.5%

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

      7. unpow2 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. associate-/r/ 2.8%

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

      2. *-inverses 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. add-sqr-sqrt 1.3%

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

      5. sqrt-unprod 29.3%

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

      6. pow2 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. unpow2 29.3%

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

      2. rem-sqrt-square 34.4%

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

   8. Simplified 34.4%

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

   if -1e-30 < (sin.f64 ky) < 1.0000000000000001e-114

   1. Initial program 84.5%

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

      1. associate-/r/ 84.5%

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

      2. +-commutative 84.5%

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

      3. unpow2 84.5%

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

      4. sqr-neg 84.5%

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

      5. sin-neg 84.5%

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

      6. sin-neg 84.5%

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

      7. unpow2 84.5%

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

      8. +-commutative 84.5%

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

   3. Simplified 99.6%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 99.6%

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

      3. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in ky around 0 47.4%

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

   if 1.0000000000000001e-114 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 56.8%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.854:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.001)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin th) 5e-11)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (/ (sin ky) (fabs (/ (sin ky) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.001) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(th) <= 5e-11) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.001) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(th) <= 5e-11) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.001:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(th) <= 5e-11:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.001)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(th) <= 5e-11)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.001)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(th) <= 5e-11)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.001], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 5e-11], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -1e-3

   1. Initial program 94.4%

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

      1. associate-/r/ 94.3%

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

      2. +-commutative 94.3%

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

      3. unpow2 94.3%

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

      4. sqr-neg 94.3%

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

      5. sin-neg 94.3%

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

      6. sin-neg 94.3%

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

      7. unpow2 94.3%

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

      8. +-commutative 94.3%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.5%

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

      3. clear-num 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 18.2%

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

   if -1e-3 < (sin.f64 th) < 5.00000000000000018e-11

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. unpow2 91.7%

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

      3. unpow2 91.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l* 89.1%

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

      3. +-commutative 89.1%

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

   6. Simplified 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-udef 16.1%

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

   8. Applied egg-rr 18.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*r/ 99.6%

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

   10. Simplified 99.6%

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

   if 5.00000000000000018e-11 < (sin.f64 th)

   1. Initial program 97.1%

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

      1. associate-/r/ 97.0%

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

      2. +-commutative 97.0%

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

      3. unpow2 97.0%

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

      4. sqr-neg 97.0%

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

      5. sin-neg 97.0%

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

      6. sin-neg 97.0%

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

      7. unpow2 97.0%

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

      8. +-commutative 97.0%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 24.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. sqrt-unprod 51.0%

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

      3. pow2 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. unpow2 51.0%

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

      2. rem-sqrt-square 52.5%

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

   8. Simplified 52.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \]

Alternative 4: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.001)
   (* (sin ky) (/ (sin th) (sin kx)))
   (if (<= (sin th) 5e-11)
     (/ th (/ (hypot (sin ky) (sin kx)) (sin ky)))
     (/ (sin ky) (fabs (/ (sin ky) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.001) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (sin(th) <= 5e-11) {
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	} else {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.001) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (Math.sin(th) <= 5e-11) {
		tmp = th / (Math.hypot(Math.sin(ky), Math.sin(kx)) / Math.sin(ky));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.001:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif math.sin(th) <= 5e-11:
		tmp = th / (math.hypot(math.sin(ky), math.sin(kx)) / math.sin(ky))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.001)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (sin(th) <= 5e-11)
		tmp = Float64(th / Float64(hypot(sin(ky), sin(kx)) / sin(ky)));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.001)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (sin(th) <= 5e-11)
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	else
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.001], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 5e-11], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -1e-3

   1. Initial program 94.4%

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

      1. associate-/r/ 94.3%

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

      2. +-commutative 94.3%

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

      3. unpow2 94.3%

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

      4. sqr-neg 94.3%

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

      5. sin-neg 94.3%

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

      6. sin-neg 94.3%

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

      7. unpow2 94.3%

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

      8. +-commutative 94.3%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.5%

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

      3. clear-num 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in ky around 0 18.2%

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

   if -1e-3 < (sin.f64 th) < 5.00000000000000018e-11

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. unpow2 91.7%

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

      3. unpow2 91.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l* 89.1%

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

      3. +-commutative 89.1%

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

   6. Simplified 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-udef 16.1%

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

   8. Applied egg-rr 18.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 89.1%

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

      4. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

   if 5.00000000000000018e-11 < (sin.f64 th)

   1. Initial program 97.1%

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

      1. associate-/r/ 97.0%

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

      2. +-commutative 97.0%

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

      3. unpow2 97.0%

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

      4. sqr-neg 97.0%

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

      5. sin-neg 97.0%

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

      6. sin-neg 97.0%

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

      7. unpow2 97.0%

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

      8. +-commutative 97.0%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 24.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. sqrt-unprod 51.0%

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

      3. pow2 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. unpow2 51.0%

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

      2. rem-sqrt-square 52.5%

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

   8. Simplified 52.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \]

Alternative 5: 74.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{th}{\frac{t_1}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin th) -0.001)
     (/ (* (sin th) ky) t_1)
     (if (<= (sin th) 5e-11)
       (/ th (/ t_1 (sin ky)))
       (/ (sin ky) (fabs (/ (sin ky) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(th) <= -0.001) {
		tmp = (sin(th) * ky) / t_1;
	} else if (sin(th) <= 5e-11) {
		tmp = th / (t_1 / sin(ky));
	} else {
		tmp = sin(ky) / fabs((sin(ky) / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(th) <= -0.001) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else if (Math.sin(th) <= 5e-11) {
		tmp = th / (t_1 / Math.sin(ky));
	} else {
		tmp = Math.sin(ky) / Math.abs((Math.sin(ky) / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(th) <= -0.001:
		tmp = (math.sin(th) * ky) / t_1
	elif math.sin(th) <= 5e-11:
		tmp = th / (t_1 / math.sin(ky))
	else:
		tmp = math.sin(ky) / math.fabs((math.sin(ky) / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(th) <= -0.001)
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	elseif (sin(th) <= 5e-11)
		tmp = Float64(th / Float64(t_1 / sin(ky)));
	else
		tmp = Float64(sin(ky) / abs(Float64(sin(ky) / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(th) <= -0.001)
		tmp = (sin(th) * ky) / t_1;
	elseif (sin(th) <= 5e-11)
		tmp = th / (t_1 / sin(ky));
	else
		tmp = sin(ky) / abs((sin(ky) / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.001], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 5e-11], N[(th / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -1e-3

   1. Initial program 94.4%

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

      1. associate-*l/ 94.3%

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

      2. +-commutative 94.3%

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

      3. unpow2 94.3%

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

      4. unpow2 94.3%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 48.6%

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

   if -1e-3 < (sin.f64 th) < 5.00000000000000018e-11

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. unpow2 91.7%

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

      3. unpow2 91.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l* 89.1%

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

      3. +-commutative 89.1%

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

   6. Simplified 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-udef 16.1%

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

   8. Applied egg-rr 18.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 89.1%

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

      4. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

   if 5.00000000000000018e-11 < (sin.f64 th)

   1. Initial program 97.1%

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

      1. associate-/r/ 97.0%

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

      2. +-commutative 97.0%

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

      3. unpow2 97.0%

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

      4. sqr-neg 97.0%

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

      5. sin-neg 97.0%

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

      6. sin-neg 97.0%

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

      7. unpow2 97.0%

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

      8. +-commutative 97.0%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 24.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. sqrt-unprod 51.0%

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

      3. pow2 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. unpow2 51.0%

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

      2. rem-sqrt-square 52.5%

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

   8. Simplified 52.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin ky}{\sin th}\right|}\\ \end{array} \]

Alternative 6: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-30)
   (fabs (sin th))
   (if (<= (sin ky) 1e-114) (* (sin ky) (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-30) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-114) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-30)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-114) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-30) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-114) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-30:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-114:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-30], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. sqr-neg 99.6%

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

      5. sin-neg 99.6%

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

      6. sin-neg 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.6%

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

      1. associate-/r/ 2.6%

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

      2. *-inverses 2.6%

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

      3. *-un-lft-identity 2.6%

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

      4. add-sqr-sqrt 1.4%

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

      5. sqrt-unprod 25.6%

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

      6. pow2 25.6%

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

   6. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 30.6%

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

   8. Simplified 30.6%

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

   if -1e-30 < (sin.f64 ky) < 1.0000000000000001e-114

   1. Initial program 84.5%

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

      1. associate-/r/ 84.5%

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

      2. +-commutative 84.5%

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

      3. unpow2 84.5%

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

      4. sqr-neg 84.5%

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

      5. sin-neg 84.5%

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

      6. sin-neg 84.5%

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

      7. unpow2 84.5%

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

      8. +-commutative 84.5%

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

   3. Simplified 99.6%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 99.6%

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

      3. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in ky around 0 47.4%

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

   if 1.0000000000000001e-114 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 56.8%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 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. +-commutative 93.9%

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

   2. unpow2 93.9%

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

   3. unpow2 93.9%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 8: 48.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-30)
   (fabs (sin th))
   (if (<= (sin ky) 1e-114) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-30) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-114) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-30)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-114) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-30) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-114) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-30:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-114:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-30], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. sqr-neg 99.6%

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

      5. sin-neg 99.6%

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

      6. sin-neg 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.6%

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

      1. associate-/r/ 2.6%

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

      2. *-inverses 2.6%

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

      3. *-un-lft-identity 2.6%

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

      4. add-sqr-sqrt 1.4%

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

      5. sqrt-unprod 25.6%

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

      6. pow2 25.6%

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

   6. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 30.6%

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

   8. Simplified 30.6%

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

   if -1e-30 < (sin.f64 ky) < 1.0000000000000001e-114

   1. Initial program 84.5%

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

      1. +-commutative 84.5%

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

      2. unpow2 84.5%

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

      3. unpow2 84.5%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 47.4%

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

   if 1.0000000000000001e-114 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 56.8%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 40.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -1e-30)
   (fabs (sin th))
   (if (<= (sin ky) 1e-114) (* th (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -1e-30) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-114) {
		tmp = th * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-1d-30)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-114) then
        tmp = th * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -1e-30) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-114) {
		tmp = th * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -1e-30:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-114:
		tmp = th * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = Float64(th * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -1e-30)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-114)
		tmp = th * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-30], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. sqr-neg 99.6%

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

      5. sin-neg 99.6%

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

      6. sin-neg 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.6%

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

      1. associate-/r/ 2.6%

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

      2. *-inverses 2.6%

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

      3. *-un-lft-identity 2.6%

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

      4. add-sqr-sqrt 1.4%

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

      5. sqrt-unprod 25.6%

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

      6. pow2 25.6%

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

   6. Applied egg-rr 25.6%

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

      1. unpow2 25.6%

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

      2. rem-sqrt-square 30.6%

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

   8. Simplified 30.6%

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

   if -1e-30 < (sin.f64 ky) < 1.0000000000000001e-114

   1. Initial program 84.5%

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

      1. +-commutative 84.5%

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

      2. unpow2 84.5%

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

      3. unpow2 84.5%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 36.1%

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

      1. *-commutative 36.1%

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

      2. associate-*l* 40.9%

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

      3. +-commutative 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in ky around 0 25.4%

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

      1. associate-/l* 30.2%

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

   9. Simplified 30.2%

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

       1. associate-/r/ 30.3%

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

   11. Applied egg-rr 30.3%

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

   if 1.0000000000000001e-114 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 56.8%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-114}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 31.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-147}:\\ \;\;\;\;\sin ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-147) (* (sin ky) (/ th kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-147) {
		tmp = sin(ky) * (th / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-147) then
        tmp = sin(ky) * (th / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-147) {
		tmp = Math.sin(ky) * (th / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-147:
		tmp = math.sin(ky) * (th / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-147)
		tmp = Float64(sin(ky) * Float64(th / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-147)
		tmp = sin(ky) * (th / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-147], N[(N[Sin[ky], $MachinePrecision] * N[(th / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 9.9999999999999997e-148

   1. Initial program 91.2%

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

      1. +-commutative 91.2%

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

      2. unpow2 91.2%

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

      3. unpow2 91.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 42.8%

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

      1. *-commutative 42.8%

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

      2. associate-*l* 45.5%

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

      3. +-commutative 45.5%

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

   6. Simplified 45.5%

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

   7. Taylor expanded in ky around 0 19.9%

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

   8. Taylor expanded in kx around 0 15.6%

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

   if 9.9999999999999997e-148 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 54.9%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-147}:\\ \;\;\;\;\sin ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 32.9% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1e-114) (* th (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-114) {
		tmp = th * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-114) then
        tmp = th * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 1e-114) {
		tmp = th * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1e-114:
		tmp = th * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1e-114)
		tmp = Float64(th * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1e-114)
		tmp = th * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 1.0000000000000001e-114

   1. Initial program 91.5%

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

      1. +-commutative 91.5%

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

      2. unpow2 91.5%

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

      3. unpow2 91.5%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 42.7%

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

      1. *-commutative 42.7%

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

      2. associate-*l* 45.4%

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

      3. +-commutative 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in ky around 0 16.4%

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

      1. associate-/l* 19.0%

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

   9. Simplified 19.0%

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

       1. associate-/r/ 19.0%

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

   11. Applied egg-rr 19.0%

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

   if 1.0000000000000001e-114 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 56.8%

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

4. Final simplification 30.5%

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

Alternative 12: 31.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1800000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 10^{-147}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1800000.0)
   (sin th)
   (if (<= ky 1e-147) (/ th (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1800000.0) {
		tmp = sin(th);
	} else if (ky <= 1e-147) {
		tmp = th / (kx / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1800000.0d0)) then
        tmp = sin(th)
    else if (ky <= 1d-147) then
        tmp = th / (kx / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1800000.0) {
		tmp = Math.sin(th);
	} else if (ky <= 1e-147) {
		tmp = th / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1800000.0:
		tmp = math.sin(th)
	elif ky <= 1e-147:
		tmp = th / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1800000.0)
		tmp = sin(th);
	elseif (ky <= 1e-147)
		tmp = Float64(th / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1800000.0)
		tmp = sin(th);
	elseif (ky <= 1e-147)
		tmp = th / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1800000.0], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1e-147], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.8e6 or 9.9999999999999997e-148 < ky

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 29.8%

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

   if -1.8e6 < ky < 9.9999999999999997e-148

   1. Initial program 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. unpow2 84.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-*l* 43.5%

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

      3. +-commutative 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in ky around 0 26.8%

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

      1. associate-/l* 31.6%

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

   9. Simplified 31.6%

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

   10. Taylor expanded in kx around 0 20.0%

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

       1. associate-/l* 25.0%

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

   12. Simplified 25.0%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1800000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 10^{-147}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 21.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{+80}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 5.7 \cdot 10^{-114}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -4.2e+80) th (if (<= ky 5.7e-114) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -4.2e+80) {
		tmp = th;
	} else if (ky <= 5.7e-114) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-4.2d+80)) then
        tmp = th
    else if (ky <= 5.7d-114) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -4.2e+80) {
		tmp = th;
	} else if (ky <= 5.7e-114) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -4.2e+80:
		tmp = th
	elif ky <= 5.7e-114:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -4.2e+80)
		tmp = th;
	elseif (ky <= 5.7e-114)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -4.2e+80)
		tmp = th;
	elseif (ky <= 5.7e-114)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -4.2e+80], th, If[LessEqual[ky, 5.7e-114], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -4.20000000000000003e80 or 5.6999999999999997e-114 < ky

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. sqr-neg 99.5%

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

      5. sin-neg 99.5%

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

      6. sin-neg 99.5%

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

      7. unpow2 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 29.7%

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

   5. Taylor expanded in th around 0 16.3%

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

   if -4.20000000000000003e80 < ky < 5.6999999999999997e-114

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

      2. unpow2 86.4%

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

      3. unpow2 86.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 42.2%

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

      3. +-commutative 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in ky around 0 25.0%

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

      1. associate-/l* 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in kx around 0 23.2%

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

       1. associate-/r/ 23.3%

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

   12. Applied egg-rr 23.3%

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

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{+80}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 5.7 \cdot 10^{-114}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 14: 21.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1.8 \cdot 10^{+83}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -1.8e+83) th (if (<= ky 8.2e-115) (* ky (/ th kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.8e+83) {
		tmp = th;
	} else if (ky <= 8.2e-115) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-1.8d+83)) then
        tmp = th
    else if (ky <= 8.2d-115) then
        tmp = ky * (th / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -1.8e+83) {
		tmp = th;
	} else if (ky <= 8.2e-115) {
		tmp = ky * (th / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -1.8e+83:
		tmp = th
	elif ky <= 8.2e-115:
		tmp = ky * (th / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -1.8e+83)
		tmp = th;
	elseif (ky <= 8.2e-115)
		tmp = Float64(ky * Float64(th / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -1.8e+83)
		tmp = th;
	elseif (ky <= 8.2e-115)
		tmp = ky * (th / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -1.8e+83], th, If[LessEqual[ky, 8.2e-115], N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.7999999999999999e83 or 8.1999999999999993e-115 < ky

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. sqr-neg 99.5%

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

      5. sin-neg 99.5%

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

      6. sin-neg 99.5%

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

      7. unpow2 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 29.7%

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

   5. Taylor expanded in th around 0 16.3%

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

   if -1.7999999999999999e83 < ky < 8.1999999999999993e-115

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

      2. unpow2 86.4%

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

      3. unpow2 86.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 42.2%

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

      3. +-commutative 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in ky around 0 25.0%

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

      1. associate-/l* 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in kx around 0 23.2%

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

       1. clear-num 23.2%

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

       2. associate-/r/ 23.2%

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

       3. clear-num 23.2%

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

   12. Applied egg-rr 23.2%

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

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1.8 \cdot 10^{+83}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 15: 21.6% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{+83}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.95 \cdot 10^{-113}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -8e+83) th (if (<= ky 1.95e-113) (/ ky (/ kx th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e+83) {
		tmp = th;
	} else if (ky <= 1.95e-113) {
		tmp = ky / (kx / th);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-8d+83)) then
        tmp = th
    else if (ky <= 1.95d-113) then
        tmp = ky / (kx / th)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e+83) {
		tmp = th;
	} else if (ky <= 1.95e-113) {
		tmp = ky / (kx / th);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -8e+83:
		tmp = th
	elif ky <= 1.95e-113:
		tmp = ky / (kx / th)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -8e+83)
		tmp = th;
	elseif (ky <= 1.95e-113)
		tmp = Float64(ky / Float64(kx / th));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -8e+83)
		tmp = th;
	elseif (ky <= 1.95e-113)
		tmp = ky / (kx / th);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -8e+83], th, If[LessEqual[ky, 1.95e-113], N[(ky / N[(kx / th), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -8.00000000000000025e83 or 1.9499999999999999e-113 < ky

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. sqr-neg 99.5%

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

      5. sin-neg 99.5%

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

      6. sin-neg 99.5%

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

      7. unpow2 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 29.7%

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

   5. Taylor expanded in th around 0 16.3%

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

   if -8.00000000000000025e83 < ky < 1.9499999999999999e-113

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

      2. unpow2 86.4%

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

      3. unpow2 86.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 42.2%

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

      3. +-commutative 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in ky around 0 25.0%

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

      1. associate-/l* 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in kx around 0 23.2%

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

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{+83}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.95 \cdot 10^{-113}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 16: 21.7% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.75 \cdot 10^{-114}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -8.6e+82) th (if (<= ky 1.75e-114) (/ th (/ kx ky)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8.6e+82) {
		tmp = th;
	} else if (ky <= 1.75e-114) {
		tmp = th / (kx / ky);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-8.6d+82)) then
        tmp = th
    else if (ky <= 1.75d-114) then
        tmp = th / (kx / ky)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8.6e+82) {
		tmp = th;
	} else if (ky <= 1.75e-114) {
		tmp = th / (kx / ky);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -8.6e+82:
		tmp = th
	elif ky <= 1.75e-114:
		tmp = th / (kx / ky)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -8.6e+82)
		tmp = th;
	elseif (ky <= 1.75e-114)
		tmp = Float64(th / Float64(kx / ky));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -8.6e+82)
		tmp = th;
	elseif (ky <= 1.75e-114)
		tmp = th / (kx / ky);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -8.6e+82], th, If[LessEqual[ky, 1.75e-114], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -8.60000000000000029e82 or 1.75e-114 < ky

   1. Initial program 99.6%

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

      1. associate-/r/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. sqr-neg 99.5%

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

      5. sin-neg 99.5%

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

      6. sin-neg 99.5%

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

      7. unpow2 99.5%

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

      8. +-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 29.7%

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

   5. Taylor expanded in th around 0 16.3%

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

   if -8.60000000000000029e82 < ky < 1.75e-114

   1. Initial program 86.4%

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

      1. +-commutative 86.4%

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

      2. unpow2 86.4%

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

      3. unpow2 86.4%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 42.2%

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

      3. +-commutative 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in ky around 0 25.0%

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

      1. associate-/l* 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in kx around 0 19.0%

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

       1. associate-/l* 23.4%

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

   12. Simplified 23.4%

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

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.75 \cdot 10^{-114}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 17: 13.7% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.9%

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

   1. associate-/r/ 93.9%

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

   2. +-commutative 93.9%

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

   3. unpow2 93.9%

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

   4. sqr-neg 93.9%

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

   5. sin-neg 93.9%

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

   6. sin-neg 93.9%

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

   7. unpow2 93.9%

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

   8. +-commutative 93.9%

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

3. Simplified 99.6%

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

4. Taylor expanded in kx around 0 21.1%

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

5. Taylor expanded in th around 0 11.0%

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

6. Final simplification 11.0%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023309 
(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)))