Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 21.2s

Alternatives: 14

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: 78.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (/
    (sin ky)
    (* (hypot (sin kx) (sin ky)) (+ (/ 1.0 th) (* th 0.16666666666666666))))
   (if (<= (sin ky) 0.04)
     (/
      (sin th)
      (* (hypot (sin ky) (sin kx)) (+ (* ky 0.16666666666666666) (/ 1.0 ky))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (sin(ky) <= 0.04) {
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) * ((ky * 0.16666666666666666) + (1.0 / ky)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (Math.sin(ky) <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(ky), Math.sin(kx)) * ((ky * 0.16666666666666666) + (1.0 / ky)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)))
	elif math.sin(ky) <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(ky), math.sin(kx)) * ((ky * 0.16666666666666666) + (1.0 / ky)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	elseif (sin(ky) <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) * Float64(Float64(ky * 0.16666666666666666) + Float64(1.0 / ky))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	elseif (sin(ky) <= 0.04)
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) * ((ky * 0.16666666666666666) + (1.0 / ky)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(ky * 0.16666666666666666), $MachinePrecision] + N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. expm1-log1p-u 99.6%

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

      4. associate-*l/ 99.6%

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

      5. expm1-udef 55.0%

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

      6. associate-*l/ 55.1%

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

      7. *-commutative 55.1%

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

   5. Applied egg-rr 55.1%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in th around 0 49.2%

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

      1. +-commutative 49.2%

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

      2. +-commutative 49.2%

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

      3. unpow2 49.2%

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

      4. unpow2 49.2%

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

      5. hypot-def 49.2%

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

      6. associate-*r* 49.2%

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

      7. +-commutative 49.2%

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

      8. unpow2 49.2%

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

      9. unpow2 49.2%

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

      10. hypot-def 49.2%

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

      11. distribute-rgt-out 49.2%

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

   10. Simplified 49.2%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 0.0400000000000000008

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. unpow2 87.8%

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

      3. unpow2 87.8%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      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 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in ky around 0 98.4%

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

   if 0.0400000000000000008 < (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 99.7%

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

   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 57.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-14)
   (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
   (if (<= (sin ky) 0.04)
     (/ (sin th) (* (hypot (sin ky) (sin kx)) (/ 1.0 ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-14) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (sin(ky) <= 0.04) {
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) * (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -5e-14) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (Math.sin(ky) <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(ky), Math.sin(kx)) * (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-14:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif math.sin(ky) <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(ky), math.sin(kx)) * (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-14)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (sin(ky) <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) * Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-14)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (sin(ky) <= 0.04)
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) * (1.0 / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-14], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5.0000000000000002e-14

   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. Step-by-step derivation

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. expm1-log1p-u 99.6%

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

      4. associate-*l/ 99.6%

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

      5. expm1-udef 54.4%

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

      6. associate-*l/ 54.4%

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

      7. *-commutative 54.4%

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

   5. Applied egg-rr 54.4%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in th around 0 49.3%

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

      1. associate-*l/ 49.3%

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

      2. +-commutative 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. hypot-def 49.3%

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

      6. *-lft-identity 49.3%

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

      7. hypot-def 49.3%

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

      8. unpow2 49.3%

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

      9. unpow2 49.3%

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

      10. +-commutative 49.3%

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

      11. unpow2 49.3%

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

      12. unpow2 49.3%

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

      13. hypot-def 49.3%

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

   10. Simplified 49.3%

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

   if -5.0000000000000002e-14 < (sin.f64 ky) < 0.0400000000000000008

   1. Initial program 87.7%

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

      1. +-commutative 87.7%

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

      2. unpow2 87.7%

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

      3. unpow2 87.7%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      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 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in ky around 0 98.2%

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

   if 0.0400000000000000008 < (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 99.7%

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

   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 57.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (/
    (sin ky)
    (* (hypot (sin kx) (sin ky)) (+ (/ 1.0 th) (* th 0.16666666666666666))))
   (if (<= (sin ky) 0.04)
     (/ (sin th) (* (hypot (sin ky) (sin kx)) (/ 1.0 ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (sin(ky) <= 0.04) {
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) * (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	} else if (Math.sin(ky) <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(ky), Math.sin(kx)) * (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)))
	elif math.sin(ky) <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(ky), math.sin(kx)) * (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666))));
	elseif (sin(ky) <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) * Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) * ((1.0 / th) + (th * 0.16666666666666666)));
	elseif (sin(ky) <= 0.04)
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) * (1.0 / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

      3. expm1-log1p-u 99.6%

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

      4. associate-*l/ 99.6%

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

      5. expm1-udef 55.0%

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

      6. associate-*l/ 55.1%

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

      7. *-commutative 55.1%

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

   5. Applied egg-rr 55.1%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in th around 0 49.2%

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

      1. +-commutative 49.2%

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

      2. +-commutative 49.2%

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

      3. unpow2 49.2%

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

      4. unpow2 49.2%

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

      5. hypot-def 49.2%

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

      6. associate-*r* 49.2%

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

      7. +-commutative 49.2%

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

      8. unpow2 49.2%

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

      9. unpow2 49.2%

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

      10. hypot-def 49.2%

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

      11. distribute-rgt-out 49.2%

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

   10. Simplified 49.2%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 0.0400000000000000008

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. unpow2 87.8%

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

      3. unpow2 87.8%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      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 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in ky around 0 98.2%

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

   if 0.0400000000000000008 < (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 99.7%

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

   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 57.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 66.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\sin ky \cdot t_1\\ \mathbf{elif}\;\sin th \leq 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|t_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin th) -0.001)
     (* (sin ky) t_1)
     (if (<= (sin th) 1e-8)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (* (sin ky) (fabs t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(th) <= -0.001) {
		tmp = sin(ky) * t_1;
	} else if (sin(th) <= 1e-8) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = sin(ky) * fabs(t_1);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(th) <= -0.001) {
		tmp = Math.sin(ky) * t_1;
	} else if (Math.sin(th) <= 1e-8) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = Math.sin(ky) * Math.abs(t_1);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(th) <= -0.001:
		tmp = math.sin(ky) * t_1
	elif math.sin(th) <= 1e-8:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = math.sin(ky) * math.fabs(t_1)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(th) <= -0.001)
		tmp = Float64(sin(ky) * t_1);
	elseif (sin(th) <= 1e-8)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(sin(ky) * abs(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(th) <= -0.001)
		tmp = sin(ky) * t_1;
	elseif (sin(th) <= 1e-8)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = sin(ky) * abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.001], N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-8], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[t$95$1], $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%

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

      3. associate-*l/ 94.4%

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

      4. +-commutative 94.4%

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

      5. unpow2 94.4%

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

      6. unpow2 94.4%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 18.2%

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

   if -1e-3 < (sin.f64 th) < 1e-8

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. unpow2 91.8%

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

      3. unpow2 91.8%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 89.2%

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

      2. *-commutative 89.2%

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

      3. expm1-log1p-u 89.2%

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

      4. associate-*l/ 99.6%

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

      5. expm1-udef 18.4%

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

      6. associate-*l/ 18.4%

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

      7. *-commutative 18.4%

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

   5. Applied egg-rr 18.4%

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

      1. expm1-def 89.2%

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

      2. expm1-log1p 89.2%

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

      3. associate-/l* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in th around 0 91.6%

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

      1. associate-*l/ 91.8%

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

      2. +-commutative 91.8%

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

      3. unpow2 91.8%

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

      4. unpow2 91.8%

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

      5. hypot-def 99.7%

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

      6. *-lft-identity 99.7%

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

      7. hypot-def 91.8%

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

      8. unpow2 91.8%

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

      9. unpow2 91.8%

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

      10. +-commutative 91.8%

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

      11. unpow2 91.8%

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

      12. unpow2 91.8%

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

      13. hypot-def 99.7%

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

   10. Simplified 99.7%

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

   if 1e-8 < (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-*l/ 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*l/ 97.0%

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

      4. +-commutative 97.0%

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

      5. unpow2 97.0%

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

      6. unpow2 97.0%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 21.7%

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

      1. add-sqr-sqrt 20.8%

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

      2. sqrt-unprod 33.5%

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

      3. pow2 33.5%

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

   6. Applied egg-rr 33.5%

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

      1. unpow2 33.5%

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

      2. rem-sqrt-square 35.1%

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

   8. Simplified 35.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. +-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 7: 42.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 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 ky around 0 30.3%

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

   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 2 regimes into one program.

4. Final simplification 38.3%

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

Alternative 8: 42.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	double tmp;
	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-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-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-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-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-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-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 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. associate-*l/ 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*l/ 91.5%

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

      4. +-commutative 91.5%

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

      5. unpow2 91.5%

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

      6. unpow2 91.5%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 30.3%

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

   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 2 regimes into one program.

4. Final simplification 38.4%

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

Alternative 9: 40.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	double tmp;
	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-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-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-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-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-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-114], N[(N[Sin[th], $MachinePrecision] * 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 ky around 0 25.7%

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

      1. associate-/l* 28.4%

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

      2. associate-/r/ 28.4%

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

   6. Simplified 28.4%

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

   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 2 regimes into one program.

4. Final simplification 37.0%

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

Alternative 10: 40.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(ky * N[(N[Sin[th], $MachinePrecision] / 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 ky around 0 25.7%

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

      1. associate-/l* 28.4%

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

      2. associate-/r/ 28.4%

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

   6. Simplified 28.4%

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

      1. associate-*l/ 25.7%

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

      2. clear-num 25.5%

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

   8. Applied egg-rr 25.5%

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

      1. associate-/r/ 25.7%

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

      2. *-commutative 25.7%

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

      3. associate-*r* 28.4%

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

      4. associate-/r/ 28.4%

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

      5. clear-num 28.4%

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

   10. Applied egg-rr 28.4%

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

   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 2 regimes into one program.

4. Final simplification 37.0%

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

Alternative 11: 34.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-114], N[(ky / N[(kx / N[Sin[th], $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 ky around 0 25.7%

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

      1. associate-/l* 28.4%

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

      2. associate-/r/ 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in kx around 0 15.8%

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

      1. associate-/l* 18.5%

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

   9. Simplified 18.5%

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

   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 2 regimes into one program.

4. Final simplification 30.2%

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

Alternative 12: 33.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1800000 \lor \neg \left(ky \leq 1.05 \cdot 10^{-114}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -1800000.0) (not (<= ky 1.05e-114)))
   (sin th)
   (* (sin th) (/ ky kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1800000.0) || !(ky <= 1.05e-114)) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / kx);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-1800000.0d0)) .or. (.not. (ky <= 1.05d-114))) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1800000.0) || !(ky <= 1.05e-114)) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -1800000.0) or not (ky <= 1.05e-114):
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -1800000.0) || !(ky <= 1.05e-114))
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -1800000.0) || ~((ky <= 1.05e-114)))
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -1800000.0], N[Not[LessEqual[ky, 1.05e-114]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -1.8e6 or 1.04999999999999996e-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. +-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 30.0%

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

   if -1.8e6 < ky < 1.04999999999999996e-114

   1. Initial program 85.4%

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

      1. +-commutative 85.4%

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

      2. unpow2 85.4%

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

      3. unpow2 85.4%

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

      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 ky around 0 42.7%

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

      1. associate-/l* 47.3%

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

      2. associate-/r/ 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in kx around 0 29.7%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1800000 \lor \neg \left(ky \leq 1.05 \cdot 10^{-114}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \]

Alternative 13: 31.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -1800000 \lor \neg \left(ky \leq 10^{-147}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -1800000.0) (not (<= ky 1e-147))) (sin th) (/ ky (/ kx th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -1800000.0) || !(ky <= 1e-147)) {
		tmp = sin(th);
	} else {
		tmp = ky / (kx / 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)) .or. (.not. (ky <= 1d-147))) then
        tmp = sin(th)
    else
        tmp = ky / (kx / th)
    end if
    code = tmp
end function

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -1800000.0) or not (ky <= 1e-147):
		tmp = math.sin(th)
	else:
		tmp = ky / (kx / th)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -1800000.0], N[Not[LessEqual[ky, 1e-147]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky / N[(kx / th), $MachinePrecision]), $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 ky around 0 42.7%

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

      1. associate-/l* 47.7%

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

      2. associate-/r/ 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in kx around 0 29.1%

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

   8. Taylor expanded in th around 0 20.0%

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

      1. associate-/l* 24.8%

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

   10. Simplified 24.8%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -1800000 \lor \neg \left(ky \leq 10^{-147}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{kx}{th}}\\ \end{array} \]

Alternative 14: 14.1% accurate, 141.8× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{\frac{kx}{th}} \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	return ky / (kx / 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 = ky / (kx / th)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(ky / N[(kx / th), $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. Taylor expanded in ky around 0 20.3%

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

   1. associate-/l* 22.1%

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

   2. associate-/r/ 22.1%

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

6. Simplified 22.1%

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

7. Taylor expanded in kx around 0 14.0%

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

8. Taylor expanded in th around 0 10.2%

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

   1. associate-/l* 12.1%

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

10. Simplified 12.1%

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

11. Final simplification 12.1%

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

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)))