Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.6% → 99.7%

Time: 15.6s

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: 94.6% 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 92.8%

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

   1. unpow2 92.8%

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

   2. sqr-neg 92.8%

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

   3. sin-neg 92.8%

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

   4. sin-neg 92.8%

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

   5. unpow2 92.8%

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

   6. associate-*l/ 90.8%

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

   7. associate-/l* 92.8%

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

   8. unpow2 92.8%

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

3. Simplified 99.6%

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

   1. associate-*r/ 96.3%

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

   2. hypot-undefine 90.8%

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

   3. unpow2 90.8%

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

   4. unpow2 90.8%

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

   5. +-commutative 90.8%

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

   6. associate-*l/ 92.8%

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

   7. *-commutative 92.8%

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

   8. clear-num 92.9%

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

   9. un-div-inv 92.9%

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

   10. +-commutative 92.9%

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

   11. unpow2 92.9%

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

   12. unpow2 92.9%

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

   13. hypot-undefine 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.71:\\ \;\;\;\;\frac{th}{\frac{t\_1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq -0.5:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{th}{t\_1 \cdot \frac{1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.008:\\ \;\;\;\;\frac{\sin th}{t\_1 \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.71)
     (/ th (/ t_1 (sin ky)))
     (if (<= (sin ky) -0.5)
       (fabs (sin th))
       (if (<= (sin ky) -0.002)
         (/ th (* t_1 (/ 1.0 (sin ky))))
         (if (<= (sin ky) 0.008) (/ (sin th) (* t_1 (/ 1.0 ky))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -0.71) {
		tmp = th / (t_1 / sin(ky));
	} else if (sin(ky) <= -0.5) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -0.002) {
		tmp = th / (t_1 * (1.0 / sin(ky)));
	} else if (sin(ky) <= 0.008) {
		tmp = sin(th) / (t_1 * (1.0 / ky));
	} else {
		tmp = 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(ky) <= -0.71) {
		tmp = th / (t_1 / Math.sin(ky));
	} else if (Math.sin(ky) <= -0.5) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -0.002) {
		tmp = th / (t_1 * (1.0 / Math.sin(ky)));
	} else if (Math.sin(ky) <= 0.008) {
		tmp = Math.sin(th) / (t_1 * (1.0 / ky));
	} else {
		tmp = 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(ky) <= -0.71:
		tmp = th / (t_1 / math.sin(ky))
	elif math.sin(ky) <= -0.5:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -0.002:
		tmp = th / (t_1 * (1.0 / math.sin(ky)))
	elif math.sin(ky) <= 0.008:
		tmp = math.sin(th) / (t_1 * (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.71)
		tmp = Float64(th / Float64(t_1 / sin(ky)));
	elseif (sin(ky) <= -0.5)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.002)
		tmp = Float64(th / Float64(t_1 * Float64(1.0 / sin(ky))));
	elseif (sin(ky) <= 0.008)
		tmp = Float64(sin(th) / Float64(t_1 * Float64(1.0 / ky)));
	else
		tmp = 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(ky) <= -0.71)
		tmp = th / (t_1 / sin(ky));
	elseif (sin(ky) <= -0.5)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.002)
		tmp = th / (t_1 * (1.0 / sin(ky)));
	elseif (sin(ky) <= 0.008)
		tmp = sin(th) / (t_1 * (1.0 / ky));
	else
		tmp = 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[ky], $MachinePrecision], -0.71], N[(th / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.5], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(th / N[(t$95$1 * N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.008], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.70999999999999996

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. hypot-undefine 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-*l/ 99.8%

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

      7. *-commutative 99.8%

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

      8. clear-num 99.7%

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

      9. un-div-inv 99.8%

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

      10. +-commutative 99.8%

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

      11. unpow2 99.8%

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

      12. unpow2 99.8%

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

      13. hypot-undefine 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 55.5%

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

   if -0.70999999999999996 < (sin.f64 ky) < -0.5

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.9%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 37.5%

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

      3. pow2 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. unpow2 37.5%

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

      2. rem-sqrt-square 45.1%

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

      3. associate-*r/ 45.1%

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

      4. associate-*l/ 45.2%

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

      5. *-inverses 45.2%

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

      6. *-lft-identity 45.2%

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

   9. Simplified 45.2%

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

   if -0.5 < (sin.f64 ky) < -2e-3

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. hypot-undefine 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-*l/ 99.6%

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

      7. *-commutative 99.6%

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

      8. clear-num 99.7%

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

      9. un-div-inv 99.7%

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

      10. +-commutative 99.7%

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

      11. unpow2 99.7%

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

      12. unpow2 99.7%

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

      13. hypot-undefine 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in th around 0 67.8%

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

   if -2e-3 < (sin.f64 ky) < 0.0080000000000000002

   1. Initial program 86.0%

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

      1. unpow2 86.0%

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

      2. sqr-neg 86.0%

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

      3. sin-neg 86.0%

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

      4. sin-neg 86.0%

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

      5. unpow2 86.0%

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

      6. associate-*l/ 81.9%

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

      7. associate-/l* 86.0%

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

      8. unpow2 86.0%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 93.0%

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

      2. hypot-undefine 81.9%

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

      3. unpow2 81.9%

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

      4. unpow2 81.9%

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

      5. +-commutative 81.9%

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

      6. associate-*l/ 86.0%

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

      7. *-commutative 86.0%

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

      8. clear-num 86.0%

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

      9. un-div-inv 86.0%

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

      10. +-commutative 86.0%

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

      11. unpow2 86.0%

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

      12. unpow2 86.0%

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

      13. hypot-undefine 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in ky around 0 99.2%

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

   if 0.0080000000000000002 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 62.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.71:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq -0.5:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}\\ \mathbf{elif}\;\sin ky \leq 0.008:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{if}\;\sin ky \leq -0.78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\sin ky \leq -0.5:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (fabs (/ (sin th) (sin ky))))))
   (if (<= (sin ky) -0.78)
     t_1
     (if (<= (sin ky) -0.5)
       (fabs (sin th))
       (if (<= (sin ky) -0.002)
         t_1
         (if (<= (sin ky) 1e-104)
           (* (sin ky) (* (sin th) (/ 1.0 (sin kx))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * fabs((sin(th) / sin(ky)));
	double tmp;
	if (sin(ky) <= -0.78) {
		tmp = t_1;
	} else if (sin(ky) <= -0.5) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -0.002) {
		tmp = t_1;
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(ky) * (sin(th) * (1.0 / 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) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) * abs((sin(th) / sin(ky)))
    if (sin(ky) <= (-0.78d0)) then
        tmp = t_1
    else if (sin(ky) <= (-0.5d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-0.002d0)) then
        tmp = t_1
    else if (sin(ky) <= 1d-104) then
        tmp = sin(ky) * (sin(th) * (1.0d0 / sin(kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	double tmp;
	if (Math.sin(ky) <= -0.78) {
		tmp = t_1;
	} else if (Math.sin(ky) <= -0.5) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -0.002) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(ky) * (Math.sin(th) * (1.0 / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	tmp = 0
	if math.sin(ky) <= -0.78:
		tmp = t_1
	elif math.sin(ky) <= -0.5:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -0.002:
		tmp = t_1
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(ky) * (math.sin(th) * (1.0 / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))))
	tmp = 0.0
	if (sin(ky) <= -0.78)
		tmp = t_1;
	elseif (sin(ky) <= -0.5)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.002)
		tmp = t_1;
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) * abs((sin(th) / sin(ky)));
	tmp = 0.0;
	if (sin(ky) <= -0.78)
		tmp = t_1;
	elseif (sin(ky) <= -0.5)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -0.002)
		tmp = t_1;
	elseif (sin(ky) <= 1e-104)
		tmp = sin(ky) * (sin(th) * (1.0 / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.78], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.5], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.78000000000000003 or -0.5 < (sin.f64 ky) < -2e-3

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 3.2%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 35.9%

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

      3. pow2 35.9%

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

   7. Applied egg-rr 35.9%

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

      1. unpow2 35.9%

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

      2. rem-sqrt-square 53.4%

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

   9. Simplified 53.4%

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

   if -0.78000000000000003 < (sin.f64 ky) < -0.5

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.8%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.3%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 31.7%

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

      3. pow2 31.7%

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

   7. Applied egg-rr 31.7%

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

      1. unpow2 31.7%

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

      2. rem-sqrt-square 37.7%

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

      3. associate-*r/ 37.7%

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

      4. associate-*l/ 37.8%

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

      5. *-inverses 37.8%

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

      6. *-lft-identity 37.8%

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

   9. Simplified 37.8%

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

   if -2e-3 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 84.4%

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

      1. unpow2 84.4%

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

      2. sqr-neg 84.4%

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

      3. sin-neg 84.4%

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

      4. sin-neg 84.4%

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

      5. unpow2 84.4%

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

      6. associate-*l/ 80.7%

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

      7. associate-/l* 84.5%

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

      8. unpow2 84.5%

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

   3. Simplified 99.7%

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

      1. clear-num 99.5%

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

      2. hypot-undefine 84.4%

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

      3. unpow2 84.4%

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

      4. unpow2 84.4%

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

      5. +-commutative 84.4%

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

      6. associate-/r/ 84.4%

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

      7. +-commutative 84.4%

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

      8. unpow2 84.4%

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

      9. unpow2 84.4%

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

      10. hypot-undefine 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in ky around 0 49.7%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.78:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq -0.5:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -0.002:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.005:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.004:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.74:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.005)
   (sin th)
   (if (<= (sin th) 0.004)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (if (<= (sin th) 0.74)
       (* (sin ky) (fabs (/ (sin th) (sin ky))))
       (* (sin ky) (fabs (/ (sin th) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.005) {
		tmp = sin(th);
	} else if (sin(th) <= 0.004) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(th) <= 0.74) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.005) {
		tmp = Math.sin(th);
	} else if (Math.sin(th) <= 0.004) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(th) <= 0.74) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.005:
		tmp = math.sin(th)
	elif math.sin(th) <= 0.004:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(th) <= 0.74:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.005)
		tmp = sin(th);
	elseif (sin(th) <= 0.004)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(th) <= 0.74)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.005)
		tmp = sin(th);
	elseif (sin(th) <= 0.004)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(th) <= 0.74)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.005], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.004], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.74], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.0050000000000000001

   1. Initial program 96.6%

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

      1. unpow2 96.6%

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

      2. sqr-neg 96.6%

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

      3. sin-neg 96.6%

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

      4. sin-neg 96.6%

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

      5. unpow2 96.6%

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

      6. associate-*l/ 96.6%

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

      7. associate-/l* 96.7%

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

      8. unpow2 96.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 26.8%

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

   if -0.0050000000000000001 < (sin.f64 th) < 0.0040000000000000001

   1. Initial program 94.0%

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

      1. unpow2 94.0%

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

      2. sqr-neg 94.0%

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

      3. sin-neg 94.0%

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

      4. sin-neg 94.0%

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

      5. unpow2 94.0%

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

      6. associate-*l/ 89.6%

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

      7. associate-/l* 94.0%

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

      8. unpow2 94.0%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 98.6%

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

   if 0.0040000000000000001 < (sin.f64 th) < 0.73999999999999999

   1. Initial program 85.7%

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

      1. unpow2 85.7%

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

      2. sqr-neg 85.7%

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

      3. sin-neg 85.7%

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

      4. sin-neg 85.7%

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

      5. unpow2 85.7%

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

      6. associate-*l/ 86.0%

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

      7. associate-/l* 85.7%

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

      8. unpow2 85.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in kx around 0 22.7%

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

      1. add-sqr-sqrt 21.4%

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

      2. sqrt-unprod 50.4%

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

      3. pow2 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. unpow2 50.4%

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

      2. rem-sqrt-square 59.4%

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

   9. Simplified 59.4%

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

   if 0.73999999999999999 < (sin.f64 th)

   1. Initial program 89.0%

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

      1. unpow2 89.0%

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

      2. sqr-neg 89.0%

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

      3. sin-neg 89.0%

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

      4. sin-neg 89.0%

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

      5. unpow2 89.0%

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

      6. associate-*l/ 88.8%

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

      7. associate-/l* 88.7%

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

      8. unpow2 88.7%

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

   3. Simplified 99.4%

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

      1. clear-num 99.3%

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

      2. hypot-undefine 88.6%

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

      3. unpow2 88.6%

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

      4. unpow2 88.6%

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

      5. +-commutative 88.6%

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

      6. associate-/r/ 88.7%

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

      7. +-commutative 88.7%

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

      8. unpow2 88.7%

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

      9. unpow2 88.7%

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

      10. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 35.7%

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

      1. add-sqr-sqrt 34.7%

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

      2. sqrt-unprod 50.9%

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

      3. pow2 50.9%

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

      4. associate-*l/ 50.9%

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

      5. *-un-lft-identity 50.9%

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

   9. Applied egg-rr 50.9%

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

       1. unpow2 50.9%

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

       2. rem-sqrt-square 51.9%

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

   11. Simplified 51.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.005:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.004:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.74:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.005:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.004:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.74:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.005)
   (sin th)
   (if (<= (sin th) 0.004)
     (/ th (/ (hypot (sin ky) (sin kx)) (sin ky)))
     (if (<= (sin th) 0.74)
       (* (sin ky) (fabs (/ (sin th) (sin ky))))
       (* (sin ky) (fabs (/ (sin th) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.005) {
		tmp = sin(th);
	} else if (sin(th) <= 0.004) {
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	} else if (sin(th) <= 0.74) {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.005) {
		tmp = Math.sin(th);
	} else if (Math.sin(th) <= 0.004) {
		tmp = th / (Math.hypot(Math.sin(ky), Math.sin(kx)) / Math.sin(ky));
	} else if (Math.sin(th) <= 0.74) {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.005:
		tmp = math.sin(th)
	elif math.sin(th) <= 0.004:
		tmp = th / (math.hypot(math.sin(ky), math.sin(kx)) / math.sin(ky))
	elif math.sin(th) <= 0.74:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.005)
		tmp = sin(th);
	elseif (sin(th) <= 0.004)
		tmp = Float64(th / Float64(hypot(sin(ky), sin(kx)) / sin(ky)));
	elseif (sin(th) <= 0.74)
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.005)
		tmp = sin(th);
	elseif (sin(th) <= 0.004)
		tmp = th / (hypot(sin(ky), sin(kx)) / sin(ky));
	elseif (sin(th) <= 0.74)
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.005], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.004], N[(th / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.74], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.0050000000000000001

   1. Initial program 96.6%

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

      1. unpow2 96.6%

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

      2. sqr-neg 96.6%

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

      3. sin-neg 96.6%

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

      4. sin-neg 96.6%

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

      5. unpow2 96.6%

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

      6. associate-*l/ 96.6%

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

      7. associate-/l* 96.7%

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

      8. unpow2 96.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 26.8%

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

   if -0.0050000000000000001 < (sin.f64 th) < 0.0040000000000000001

   1. Initial program 94.0%

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

      1. unpow2 94.0%

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

      2. sqr-neg 94.0%

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

      3. sin-neg 94.0%

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

      4. sin-neg 94.0%

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

      5. unpow2 94.0%

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

      6. associate-*l/ 89.6%

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

      7. associate-/l* 94.0%

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

      8. unpow2 94.0%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 92.7%

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

      2. hypot-undefine 89.6%

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

      3. unpow2 89.6%

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

      4. unpow2 89.6%

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

      5. +-commutative 89.6%

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

      6. associate-*l/ 94.0%

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

      7. *-commutative 94.0%

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

      8. clear-num 94.1%

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

      9. un-div-inv 94.1%

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

      10. +-commutative 94.1%

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

      11. unpow2 94.1%

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

      12. unpow2 94.1%

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

      13. hypot-undefine 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in th around 0 98.7%

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

   if 0.0040000000000000001 < (sin.f64 th) < 0.73999999999999999

   1. Initial program 85.7%

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

      1. unpow2 85.7%

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

      2. sqr-neg 85.7%

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

      3. sin-neg 85.7%

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

      4. sin-neg 85.7%

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

      5. unpow2 85.7%

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

      6. associate-*l/ 86.0%

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

      7. associate-/l* 85.7%

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

      8. unpow2 85.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in kx around 0 22.7%

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

      1. add-sqr-sqrt 21.4%

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

      2. sqrt-unprod 50.4%

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

      3. pow2 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. unpow2 50.4%

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

      2. rem-sqrt-square 59.4%

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

   9. Simplified 59.4%

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

   if 0.73999999999999999 < (sin.f64 th)

   1. Initial program 89.0%

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

      1. unpow2 89.0%

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

      2. sqr-neg 89.0%

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

      3. sin-neg 89.0%

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

      4. sin-neg 89.0%

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

      5. unpow2 89.0%

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

      6. associate-*l/ 88.8%

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

      7. associate-/l* 88.7%

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

      8. unpow2 88.7%

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

   3. Simplified 99.4%

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

      1. clear-num 99.3%

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

      2. hypot-undefine 88.6%

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

      3. unpow2 88.6%

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

      4. unpow2 88.6%

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

      5. +-commutative 88.6%

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

      6. associate-/r/ 88.7%

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

      7. +-commutative 88.7%

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

      8. unpow2 88.7%

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

      9. unpow2 88.7%

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

      10. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 35.7%

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

      1. add-sqr-sqrt 34.7%

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

      2. sqrt-unprod 50.9%

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

      3. pow2 50.9%

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

      4. associate-*l/ 50.9%

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

      5. *-un-lft-identity 50.9%

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

   9. Applied egg-rr 50.9%

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

       1. unpow2 50.9%

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

       2. rem-sqrt-square 51.9%

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

   11. Simplified 51.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.005:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.004:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.74:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.45:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.45)
   (fabs (sin th))
   (if (<= (sin ky) 1e-104)
     (* (sin ky) (* (sin th) (/ 1.0 (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.45) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(ky) * (sin(th) * (1.0 / 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) <= (-0.45d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-104) then
        tmp = sin(ky) * (sin(th) * (1.0d0 / 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) <= -0.45) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(ky) * (Math.sin(th) * (1.0 / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.45:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(ky) * (math.sin(th) * (1.0 / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.45)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.45)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = sin(ky) * (sin(th) * (1.0 / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.45], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.450000000000000011

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.8%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 24.6%

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

      3. pow2 24.6%

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

   7. Applied egg-rr 24.6%

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

      1. unpow2 24.6%

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

      2. rem-sqrt-square 26.6%

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

      3. associate-*r/ 26.6%

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

      4. associate-*l/ 26.6%

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

      5. *-inverses 26.6%

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

      6. *-lft-identity 26.6%

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

   9. Simplified 26.6%

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

   if -0.450000000000000011 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 83.4%

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

      7. associate-/l* 86.6%

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

      8. unpow2 86.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. hypot-undefine 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. +-commutative 86.5%

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

      6. associate-/r/ 86.5%

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

      7. +-commutative 86.5%

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

      8. unpow2 86.5%

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

      9. unpow2 86.5%

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

      10. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 43.3%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.0%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.45:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.45:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.45)
   (fabs (sin th))
   (if (<= (sin ky) 1e-104) (* (sin ky) (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.45) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-104) {
		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) <= (-0.45d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-104) 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) <= -0.45) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-104) {
		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) <= -0.45:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-104:
		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) <= -0.45)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		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) <= -0.45)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		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], -0.45], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], 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) < -0.450000000000000011

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.8%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-unprod 24.6%

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

      3. pow2 24.6%

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

   7. Applied egg-rr 24.6%

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

      1. unpow2 24.6%

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

      2. rem-sqrt-square 26.6%

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

      3. associate-*r/ 26.6%

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

      4. associate-*l/ 26.6%

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

      5. *-inverses 26.6%

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

      6. *-lft-identity 26.6%

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

   9. Simplified 26.6%

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

   if -0.450000000000000011 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 86.6%

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

      1. unpow2 86.6%

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

      2. sqr-neg 86.6%

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

      3. sin-neg 86.6%

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

      4. sin-neg 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*l/ 83.4%

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

      7. associate-/l* 86.6%

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

      8. unpow2 86.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 43.3%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.0%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.45:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. unpow2 92.8%

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

   2. sqr-neg 92.8%

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

   3. sin-neg 92.8%

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

   4. sin-neg 92.8%

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

   5. unpow2 92.8%

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

   6. associate-*l/ 90.8%

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

   7. associate-/l* 92.8%

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

   8. unpow2 92.8%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 9: 39.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;ky \cdot \frac{\mathsf{log1p}\left(th\right)}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (fabs (sin th))
   (if (<= (sin ky) 5e-151) (* ky (/ (log1p th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-151) {
		tmp = ky * (log1p(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.002) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-151) {
		tmp = ky * (Math.log1p(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.002:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-151:
		tmp = ky * (math.log1p(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-151)
		tmp = Float64(ky * Float64(log1p(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-151], N[(ky * N[(N[Log[1 + th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -2e-3

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 24.7%

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

      3. pow2 24.7%

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

   7. Applied egg-rr 24.7%

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

      1. unpow2 24.7%

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

      2. rem-sqrt-square 27.7%

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

      3. associate-*r/ 27.6%

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

      4. associate-*l/ 27.7%

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

      5. *-inverses 27.7%

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

      6. *-lft-identity 27.7%

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

   9. Simplified 27.7%

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

   if -2e-3 < (sin.f64 ky) < 5.00000000000000003e-151

   1. Initial program 83.1%

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

      1. unpow2 83.1%

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

      2. sqr-neg 83.1%

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

      3. sin-neg 83.1%

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

      4. sin-neg 83.1%

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

      5. unpow2 83.1%

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

      6. associate-*l/ 79.1%

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

      7. associate-/l* 83.2%

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

      8. unpow2 83.2%

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

   3. Simplified 99.7%

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

      1. add-log-exp 24.9%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   6. Applied egg-rr 75.8%

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

   7. Taylor expanded in ky around 0 42.1%

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

   8. Taylor expanded in th around 0 23.5%

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

      1. +-commutative 23.0%

         \[\leadsto \log \color{blue}{\left(th + 1\right)} \]

   10. Simplified 23.5%

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

   11. Taylor expanded in ky around 0 23.5%

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

       1. log1p-define 30.3%

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

       2. associate-/l* 32.0%

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

   13. Simplified 32.0%

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

   if 5.00000000000000003e-151 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 58.9%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;ky \cdot \frac{\mathsf{log1p}\left(th\right)}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (fabs (sin th))
   (if (<= (sin ky) 1e-104) (* ky (/ (sin th) (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-104) {
		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) <= (-0.002d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-104) 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) <= -0.002) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-104) {
		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) <= -0.002:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-104:
		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) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		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) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		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], -0.002], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(ky * 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) < -2e-3

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 24.7%

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

      3. pow2 24.7%

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

   7. Applied egg-rr 24.7%

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

      1. unpow2 24.7%

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

      2. rem-sqrt-square 27.7%

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

      3. associate-*r/ 27.6%

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

      4. associate-*l/ 27.7%

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

      5. *-inverses 27.7%

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

      6. *-lft-identity 27.7%

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

   9. Simplified 27.7%

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

   if -2e-3 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 84.4%

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

      1. unpow2 84.4%

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

      2. sqr-neg 84.4%

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

      3. sin-neg 84.4%

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

      4. sin-neg 84.4%

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

      5. unpow2 84.4%

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

      6. associate-*l/ 80.7%

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

      7. associate-/l* 84.5%

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

      8. unpow2 84.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 48.1%

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

      1. associate-/l* 49.7%

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

   7. Simplified 49.7%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 59.0%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (fabs (sin th))
   (if (<= (sin ky) 5e-151) (* ky (/ th (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-151) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.002d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-151) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.002) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-151) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.002:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-151:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-151)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-151)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-151], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -2e-3

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.7%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 3.0%

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

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 24.7%

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

      3. pow2 24.7%

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

   7. Applied egg-rr 24.7%

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

      1. unpow2 24.7%

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

      2. rem-sqrt-square 27.7%

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

      3. associate-*r/ 27.6%

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

      4. associate-*l/ 27.7%

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

      5. *-inverses 27.7%

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

      6. *-lft-identity 27.7%

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

   9. Simplified 27.7%

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

   if -2e-3 < (sin.f64 ky) < 5.00000000000000003e-151

   1. Initial program 83.1%

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

      1. unpow2 83.1%

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

      2. sqr-neg 83.1%

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

      3. sin-neg 83.1%

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

      4. sin-neg 83.1%

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

      5. unpow2 83.1%

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

      6. associate-*l/ 79.1%

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

      7. associate-/l* 83.2%

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

      8. unpow2 83.2%

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

   3. Simplified 99.7%

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

      1. clear-num 99.5%

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

      2. hypot-undefine 83.1%

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

      3. unpow2 83.1%

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

      4. unpow2 83.1%

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

      5. +-commutative 83.1%

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

      6. associate-/r/ 83.1%

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

      7. +-commutative 83.1%

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

      8. unpow2 83.1%

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

      9. unpow2 83.1%

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

      10. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 50.9%

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

   8. Taylor expanded in th around 0 32.5%

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

   9. Taylor expanded in ky around 0 31.0%

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

       1. associate-/l* 32.5%

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

   11. Simplified 32.5%

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

   if 5.00000000000000003e-151 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 58.9%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;th \cdot \frac{\sin ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 5e-151) (* th (/ (sin ky) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 5e-151) {
		tmp = th * (sin(ky) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 5d-151) then
        tmp = th * (sin(ky) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 5e-151) {
		tmp = th * (Math.sin(ky) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 5e-151:
		tmp = th * (math.sin(ky) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 5e-151)
		tmp = Float64(th * Float64(sin(ky) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 5e-151)
		tmp = th * (sin(ky) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-151], N[(th * N[(N[Sin[ky], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 5.00000000000000003e-151

   1. Initial program 89.1%

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

      1. unpow2 89.1%

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

      2. sqr-neg 89.1%

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

      3. sin-neg 89.1%

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

      4. sin-neg 89.1%

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

      5. unpow2 89.1%

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

      6. associate-*l/ 86.5%

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

      7. associate-/l* 89.0%

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

      8. unpow2 89.0%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. hypot-undefine 88.9%

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

      3. unpow2 88.9%

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

      4. unpow2 88.9%

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

      5. +-commutative 88.9%

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

      6. associate-/r/ 88.9%

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

      7. +-commutative 88.9%

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

      8. unpow2 88.9%

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

      9. unpow2 88.9%

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

      10. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 34.8%

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

   8. Taylor expanded in th around 0 22.5%

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

   9. Taylor expanded in kx around 0 18.9%

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

       1. associate-/l* 19.9%

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

   11. Simplified 19.9%

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

   if 5.00000000000000003e-151 < (sin.f64 ky)

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 58.9%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;th \cdot \frac{\sin ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 25.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.6 \cdot 10^{-151}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.6e-151) (* ky (/ th (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.6e-151) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 4.6d-151) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.6e-151) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.6e-151:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.6e-151)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.6e-151)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.6e-151], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.59999999999999992e-151

   1. Initial program 89.5%

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

      1. unpow2 89.5%

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

      2. sqr-neg 89.5%

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

      3. sin-neg 89.5%

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

      4. sin-neg 89.5%

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

      5. unpow2 89.5%

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

      6. associate-*l/ 87.1%

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

      7. associate-/l* 89.5%

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

      8. unpow2 89.5%

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

   3. Simplified 99.6%

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

      1. clear-num 99.4%

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

      2. hypot-undefine 89.4%

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

      3. unpow2 89.4%

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

      4. unpow2 89.4%

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

      5. +-commutative 89.4%

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

      6. associate-/r/ 89.4%

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

      7. +-commutative 89.4%

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

      8. unpow2 89.4%

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

      9. unpow2 89.4%

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

      10. hypot-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ky around 0 34.1%

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

   8. Taylor expanded in th around 0 22.4%

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

   9. Taylor expanded in ky around 0 20.0%

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

       1. associate-/l* 20.9%

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

   11. Simplified 20.9%

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

   if 4.59999999999999992e-151 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 98.5%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 42.9%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.6 \cdot 10^{-151}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\log \left(th + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 7.8e-7) (sin th) (log (+ th 1.0))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 7.8e-7) {
		tmp = sin(th);
	} else {
		tmp = log((th + 1.0));
	}
	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 (kx <= 7.8d-7) then
        tmp = sin(th)
    else
        tmp = log((th + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 7.8e-7) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.log((th + 1.0));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 7.8e-7:
		tmp = math.sin(th)
	else:
		tmp = math.log((th + 1.0))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 7.8e-7)
		tmp = sin(th);
	else
		tmp = log(Float64(th + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 7.8e-7)
		tmp = sin(th);
	else
		tmp = log((th + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 7.8e-7], N[Sin[th], $MachinePrecision], N[Log[N[(th + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.80000000000000049e-7

   1. Initial program 91.1%

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

      1. unpow2 91.1%

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

      2. sqr-neg 91.1%

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

      3. sin-neg 91.1%

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

      4. sin-neg 91.1%

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

      5. unpow2 91.1%

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

      6. associate-*l/ 88.5%

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

      7. associate-/l* 91.0%

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

      8. unpow2 91.0%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 29.0%

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

   if 7.80000000000000049e-7 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.4%

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

      8. unpow2 99.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in kx around 0 9.2%

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

      1. add-log-exp 20.9%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   7. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   8. Taylor expanded in th around 0 18.6%

      \[\leadsto \log \color{blue}{\left(1 + th\right)} \]
   9. Step-by-step derivation

      1. +-commutative 18.6%

         \[\leadsto \log \color{blue}{\left(th + 1\right)} \]

   10. Simplified 18.6%

       \[\leadsto \log \color{blue}{\left(th + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\log \left(th + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 14.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.75:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\log th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= th 1.75) th (log th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.75) {
		tmp = th;
	} else {
		tmp = log(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.75d0) then
        tmp = th
    else
        tmp = log(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.75) {
		tmp = th;
	} else {
		tmp = Math.log(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 1.75:
		tmp = th
	else:
		tmp = math.log(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 1.75)
		tmp = th;
	else
		tmp = log(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 1.75)
		tmp = th;
	else
		tmp = log(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 1.75], th, N[Log[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.75

   1. Initial program 91.5%

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

      1. unpow2 91.5%

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

      2. sqr-neg 91.5%

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

      3. sin-neg 91.5%

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

      4. sin-neg 91.5%

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

      5. unpow2 91.5%

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

      6. associate-*l/ 88.8%

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

      7. associate-/l* 91.5%

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

      8. unpow2 91.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 21.1%

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

   6. Taylor expanded in th around 0 16.0%

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

   if 1.75 < th

   1. Initial program 96.7%

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

      1. unpow2 96.7%

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

      2. sqr-neg 96.7%

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

      3. sin-neg 96.7%

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

      4. sin-neg 96.7%

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

      5. unpow2 96.7%

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

      6. associate-*l/ 96.7%

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

      7. associate-/l* 96.6%

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

      8. unpow2 96.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 35.6%

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

      1. add-log-exp 35.3%

         \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   7. Applied egg-rr 35.3%

      \[\leadsto \color{blue}{\log \left(e^{\sin th}\right)} \]

   8. Taylor expanded in th around 0 6.0%

      \[\leadsto \log \color{blue}{\left(1 + th\right)} \]
   9. Step-by-step derivation

      1. +-commutative 6.0%

         \[\leadsto \log \color{blue}{\left(th + 1\right)} \]

   10. Simplified 6.0%

       \[\leadsto \log \color{blue}{\left(th + 1\right)} \]

   11. Taylor expanded in th around inf 6.0%

       \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{th}\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 6.0%

          \[\leadsto \color{blue}{-\log \left(\frac{1}{th}\right)} \]

       2. log-rec 6.0%

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

       3. remove-double-neg 6.0%

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

   13. Simplified 6.0%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.75:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\log th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

   1. unpow2 92.8%

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

   2. sqr-neg 92.8%

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

   3. sin-neg 92.8%

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

   4. sin-neg 92.8%

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

   5. unpow2 92.8%

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

   6. associate-*l/ 90.8%

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

   7. associate-/l* 92.8%

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

   8. unpow2 92.8%

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

3. Simplified 99.6%

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

5. Taylor expanded in kx around 0 24.8%

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

6. Final simplification 24.8%

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

Alternative 17: 13.4% 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 92.8%

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

   1. unpow2 92.8%

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

   2. sqr-neg 92.8%

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

   3. sin-neg 92.8%

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

   4. sin-neg 92.8%

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

   5. unpow2 92.8%

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

   6. associate-*l/ 90.8%

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

   7. associate-/l* 92.8%

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

   8. unpow2 92.8%

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

3. Simplified 99.6%

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

5. Taylor expanded in kx around 0 24.8%

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

6. Taylor expanded in th around 0 12.6%

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

7. Final simplification 12.6%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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)))