Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 20.3s

Alternatives: 23

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. +-commutative 93.4%

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

   2. unpow2 93.4%

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

   3. unpow2 93.4%

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

   4. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 47.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.16:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-30}\right) \land \sin ky \leq 10^{-11}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.16)
   (- th)
   (if (or (<= (sin ky) 4e-93)
           (and (not (<= (sin ky) 5e-30)) (<= (sin ky) 1e-11)))
     (/ (sin ky) (/ (sin kx) (sin th)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.16) {
		tmp = -th;
	} else if ((sin(ky) <= 4e-93) || (!(sin(ky) <= 5e-30) && (sin(ky) <= 1e-11))) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.16d0)) then
        tmp = -th
    else if ((sin(ky) <= 4d-93) .or. (.not. (sin(ky) <= 5d-30)) .and. (sin(ky) <= 1d-11)) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.16) {
		tmp = -th;
	} else if ((Math.sin(ky) <= 4e-93) || (!(Math.sin(ky) <= 5e-30) && (Math.sin(ky) <= 1e-11))) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.16:
		tmp = -th
	elif (math.sin(ky) <= 4e-93) or (not (math.sin(ky) <= 5e-30) and (math.sin(ky) <= 1e-11)):
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.16)
		tmp = Float64(-th);
	elseif ((sin(ky) <= 4e-93) || (!(sin(ky) <= 5e-30) && (sin(ky) <= 1e-11)))
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.16)
		tmp = -th;
	elseif ((sin(ky) <= 4e-93) || (~((sin(ky) <= 5e-30)) && (sin(ky) <= 1e-11)))
		tmp = sin(ky) / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.16], (-th), If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 4e-93], And[N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 5e-30]], $MachinePrecision], LessEqual[N[Sin[ky], $MachinePrecision], 1e-11]]], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.160000000000000003

   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/ 99.5%

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

      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 99.5%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 3.4%

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

   7. Applied egg-rr 3.4%

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

      1. expm1-define 2.6%

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

      2. associate-*r/ 2.6%

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

      3. associate-*l/ 2.6%

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

      4. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 3.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 3.4%

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

       3. rem-exp-log 3.4%

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

       4. +-commutative 3.4%

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

   11. Applied egg-rr 3.4%

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

   12. Taylor expanded in th around 0 4.3%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 3.2%

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

       2. expm1-define 3.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 2.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 1.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 1.6%

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

       6. sqrt-unprod 10.4%

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

       7. pow2 10.4%

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

   14. Applied egg-rr 10.4%

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

   15. Taylor expanded in th around -inf 35.6%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 35.6%

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

   17. Simplified 35.6%

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

   if -0.160000000000000003 < (sin.f64 ky) < 3.9999999999999996e-93 or 4.99999999999999972e-30 < (sin.f64 ky) < 9.99999999999999939e-12

   1. Initial program 84.5%

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

      1. unpow2 84.5%

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

      2. sqr-neg 84.5%

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

      3. sin-neg 84.5%

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

      4. sin-neg 84.5%

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

      5. unpow2 84.5%

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

      6. associate-*l/ 81.2%

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

      7. associate-/l* 84.4%

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

      8. unpow2 84.4%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   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. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in ky around 0 50.5%

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

   if 3.9999999999999996e-93 < (sin.f64 ky) < 4.99999999999999972e-30 or 9.99999999999999939e-12 < (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* 99.7%

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

      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 57.7%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.16:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-30}\right) \land \sin ky \leq 10^{-11}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin kx) -0.0638)
     (* (/ (sin ky) t_1) th)
     (if (<= (sin kx) 1e-7)
       (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
       (if (<= (sin kx) 0.42)
         (/ (* (sin ky) th) t_1)
         (/ (sin ky) (* (sin kx) (/ 1.0 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(kx) <= -0.0638) {
		tmp = (sin(ky) / t_1) * th;
	} else if (sin(kx) <= 1e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (sin(kx) <= 0.42) {
		tmp = (sin(ky) * th) / t_1;
	} else {
		tmp = sin(ky) / (sin(kx) * (1.0 / 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(kx) <= -0.0638) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (Math.sin(kx) <= 1e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(kx) <= 0.42) {
		tmp = (Math.sin(ky) * th) / t_1;
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) * (1.0 / 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(kx) <= -0.0638:
		tmp = (math.sin(ky) / t_1) * th
	elif math.sin(kx) <= 1e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif math.sin(kx) <= 0.42:
		tmp = (math.sin(ky) * th) / t_1
	else:
		tmp = math.sin(ky) / (math.sin(kx) * (1.0 / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.0638)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (sin(kx) <= 1e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (sin(kx) <= 0.42)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) * Float64(1.0 / 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(kx) <= -0.0638)
		tmp = (sin(ky) / t_1) * th;
	elseif (sin(kx) <= 1e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (sin(kx) <= 0.42)
		tmp = (sin(ky) * th) / t_1;
	else
		tmp = sin(ky) / (sin(kx) * (1.0 / 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[kx], $MachinePrecision], -0.0638], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.42], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] * N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.063799999999999996

   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/ 99.3%

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

      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 99.4%

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

      1. associate-*r/ 99.4%

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

      2. clear-num 97.7%

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

      3. *-commutative 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in th around 0 49.4%

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

      1. *-commutative 49.4%

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

   9. Simplified 49.4%

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

       1. associate-/r* 49.4%

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

       2. associate-/r/ 51.2%

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

       3. clear-num 51.2%

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

   11. Applied egg-rr 51.2%

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

   if -0.063799999999999996 < (sin.f64 kx) < 9.9999999999999995e-8

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. unpow2 86.5%

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

      3. unpow2 86.5%

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

      4. hypot-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in kx around 0 98.3%

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

   if 9.9999999999999995e-8 < (sin.f64 kx) < 0.419999999999999984

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sqr-neg 99.3%

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

      3. sin-neg 99.3%

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

      4. sin-neg 99.3%

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

      5. unpow2 99.3%

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

      6. associate-*l/ 99.1%

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

      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 99.6%

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

      1. associate-*r/ 99.3%

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

      2. clear-num 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in th around 0 54.4%

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

      1. *-commutative 54.4%

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

   9. Simplified 54.4%

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

       1. associate-/r* 54.6%

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

       2. associate-/r/ 54.6%

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

       3. clear-num 54.7%

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

   11. Applied egg-rr 54.7%

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

       1. associate-*l/ 54.5%

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

   13. Applied egg-rr 54.5%

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

   if 0.419999999999999984 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      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 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   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. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in ky around 0 61.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx} \cdot \frac{1}{\sin th}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;\sin ky \cdot \frac{th}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin kx) -0.0638)
     (* (/ (sin ky) t_1) th)
     (if (<= (sin kx) 1e-7)
       (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
       (if (<= (sin kx) 0.42)
         (* (sin ky) (/ th t_1))
         (/ (sin ky) (* (sin kx) (/ 1.0 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(kx) <= -0.0638) {
		tmp = (sin(ky) / t_1) * th;
	} else if (sin(kx) <= 1e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (sin(kx) <= 0.42) {
		tmp = sin(ky) * (th / t_1);
	} else {
		tmp = sin(ky) / (sin(kx) * (1.0 / 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(kx) <= -0.0638) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (Math.sin(kx) <= 1e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(kx) <= 0.42) {
		tmp = Math.sin(ky) * (th / t_1);
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) * (1.0 / 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(kx) <= -0.0638:
		tmp = (math.sin(ky) / t_1) * th
	elif math.sin(kx) <= 1e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif math.sin(kx) <= 0.42:
		tmp = math.sin(ky) * (th / t_1)
	else:
		tmp = math.sin(ky) / (math.sin(kx) * (1.0 / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.0638)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (sin(kx) <= 1e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (sin(kx) <= 0.42)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) * Float64(1.0 / 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(kx) <= -0.0638)
		tmp = (sin(ky) / t_1) * th;
	elseif (sin(kx) <= 1e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (sin(kx) <= 0.42)
		tmp = sin(ky) * (th / t_1);
	else
		tmp = sin(ky) / (sin(kx) * (1.0 / 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[kx], $MachinePrecision], -0.0638], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.42], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] * N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.063799999999999996

   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/ 99.3%

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

      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 99.4%

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

      1. associate-*r/ 99.4%

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

      2. clear-num 97.7%

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

      3. *-commutative 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in th around 0 49.4%

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

      1. *-commutative 49.4%

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

   9. Simplified 49.4%

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

       1. associate-/r* 49.4%

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

       2. associate-/r/ 51.2%

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

       3. clear-num 51.2%

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

   11. Applied egg-rr 51.2%

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

   if -0.063799999999999996 < (sin.f64 kx) < 9.9999999999999995e-8

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. unpow2 86.5%

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

      3. unpow2 86.5%

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

      4. hypot-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in kx around 0 98.3%

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

   if 9.9999999999999995e-8 < (sin.f64 kx) < 0.419999999999999984

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. sqr-neg 99.3%

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

      3. sin-neg 99.3%

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

      4. sin-neg 99.3%

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

      5. unpow2 99.3%

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

      6. associate-*l/ 99.1%

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

      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 99.6%

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

      1. associate-*r/ 99.3%

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

      2. clear-num 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in th around 0 54.4%

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

      1. *-commutative 54.4%

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

   9. Simplified 54.4%

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

       1. *-un-lft-identity 54.4%

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

       2. associate-/r* 54.6%

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

       3. associate-/r/ 54.6%

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

       4. clear-num 54.7%

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

   11. Applied egg-rr 54.7%

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

       1. *-lft-identity 54.7%

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

       2. associate-*l/ 54.5%

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

       3. associate-*r/ 54.7%

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

   13. Simplified 54.7%

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

   if 0.419999999999999984 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      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 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   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. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in ky around 0 61.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx} \cdot \frac{1}{\sin th}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ th (hypot (sin ky) (sin kx))))))
   (if (<= (sin kx) -0.0638)
     t_1
     (if (<= (sin kx) 1e-7)
       (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
       (if (<= (sin kx) 0.42)
         t_1
         (/ (sin ky) (* (sin kx) (/ 1.0 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	double tmp;
	if (sin(kx) <= -0.0638) {
		tmp = t_1;
	} else if (sin(kx) <= 1e-7) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (sin(kx) <= 0.42) {
		tmp = t_1;
	} else {
		tmp = sin(ky) / (sin(kx) * (1.0 / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	double tmp;
	if (Math.sin(kx) <= -0.0638) {
		tmp = t_1;
	} else if (Math.sin(kx) <= 1e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(kx) <= 0.42) {
		tmp = t_1;
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) * (1.0 / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	tmp = 0
	if math.sin(kx) <= -0.0638:
		tmp = t_1
	elif math.sin(kx) <= 1e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif math.sin(kx) <= 0.42:
		tmp = t_1
	else:
		tmp = math.sin(ky) / (math.sin(kx) * (1.0 / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))))
	tmp = 0.0
	if (sin(kx) <= -0.0638)
		tmp = t_1;
	elseif (sin(kx) <= 1e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (sin(kx) <= 0.42)
		tmp = t_1;
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) * Float64(1.0 / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	tmp = 0.0;
	if (sin(kx) <= -0.0638)
		tmp = t_1;
	elseif (sin(kx) <= 1e-7)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	elseif (sin(kx) <= 0.42)
		tmp = t_1;
	else
		tmp = sin(ky) / (sin(kx) * (1.0 / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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[kx], $MachinePrecision], -0.0638], t$95$1, If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.42], t$95$1, N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] * N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.063799999999999996 or 9.9999999999999995e-8 < (sin.f64 kx) < 0.419999999999999984

   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/ 99.3%

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

      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/ 99.3%

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

      2. clear-num 98.0%

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

      3. *-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in th around 0 50.5%

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

      1. *-commutative 50.5%

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

   9. Simplified 50.5%

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

       1. *-un-lft-identity 50.5%

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

       2. associate-/r* 50.5%

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

       3. associate-/r/ 51.9%

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

       4. clear-num 52.0%

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

   11. Applied egg-rr 52.0%

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

       1. *-lft-identity 52.0%

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

       2. associate-*l/ 51.8%

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

       3. associate-*r/ 51.9%

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

   13. Simplified 51.9%

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

   if -0.063799999999999996 < (sin.f64 kx) < 9.9999999999999995e-8

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. unpow2 86.5%

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

      3. unpow2 86.5%

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

      4. hypot-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in kx around 0 98.3%

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

   if 0.419999999999999984 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      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 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   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. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in ky around 0 61.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx} \cdot \frac{1}{\sin th}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;\sin kx \leq -0.0638:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\sin kx \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 0.42:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx \cdot \frac{1}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ th (hypot (sin ky) (sin kx))))))
   (if (<= (sin kx) -0.0638)
     t_1
     (if (<= (sin kx) 1e-7)
       (* (sin ky) (/ (sin th) (hypot (sin ky) kx)))
       (if (<= (sin kx) 0.42)
         t_1
         (/ (sin ky) (* (sin kx) (/ 1.0 (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	double tmp;
	if (sin(kx) <= -0.0638) {
		tmp = t_1;
	} else if (sin(kx) <= 1e-7) {
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	} else if (sin(kx) <= 0.42) {
		tmp = t_1;
	} else {
		tmp = sin(ky) / (sin(kx) * (1.0 / sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	double tmp;
	if (Math.sin(kx) <= -0.0638) {
		tmp = t_1;
	} else if (Math.sin(kx) <= 1e-7) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), kx));
	} else if (Math.sin(kx) <= 0.42) {
		tmp = t_1;
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) * (1.0 / Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	tmp = 0
	if math.sin(kx) <= -0.0638:
		tmp = t_1
	elif math.sin(kx) <= 1e-7:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), kx))
	elif math.sin(kx) <= 0.42:
		tmp = t_1
	else:
		tmp = math.sin(ky) / (math.sin(kx) * (1.0 / math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))))
	tmp = 0.0
	if (sin(kx) <= -0.0638)
		tmp = t_1;
	elseif (sin(kx) <= 1e-7)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), kx)));
	elseif (sin(kx) <= 0.42)
		tmp = t_1;
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) * Float64(1.0 / sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	tmp = 0.0;
	if (sin(kx) <= -0.0638)
		tmp = t_1;
	elseif (sin(kx) <= 1e-7)
		tmp = sin(ky) * (sin(th) / hypot(sin(ky), kx));
	elseif (sin(kx) <= 0.42)
		tmp = t_1;
	else
		tmp = sin(ky) / (sin(kx) * (1.0 / sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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[kx], $MachinePrecision], -0.0638], t$95$1, If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-7], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.42], t$95$1, N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] * N[(1.0 / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.063799999999999996 or 9.9999999999999995e-8 < (sin.f64 kx) < 0.419999999999999984

   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/ 99.3%

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

      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/ 99.3%

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

      2. clear-num 98.0%

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

      3. *-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in th around 0 50.5%

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

      1. *-commutative 50.5%

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

   9. Simplified 50.5%

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

       1. *-un-lft-identity 50.5%

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

       2. associate-/r* 50.5%

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

       3. associate-/r/ 51.9%

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

       4. clear-num 52.0%

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

   11. Applied egg-rr 52.0%

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

       1. *-lft-identity 52.0%

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

       2. associate-*l/ 51.8%

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

       3. associate-*r/ 51.9%

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

   13. Simplified 51.9%

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

   if -0.063799999999999996 < (sin.f64 kx) < 9.9999999999999995e-8

   1. Initial program 86.5%

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

      1. unpow2 86.5%

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

      2. sqr-neg 86.5%

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

      3. sin-neg 86.5%

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

      4. sin-neg 86.5%

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

      5. unpow2 86.5%

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

      6. associate-*l/ 83.0%

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

      7. associate-/l* 86.5%

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

      8. unpow2 86.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 98.2%

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

   if 0.419999999999999984 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      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 99.4%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   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. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-inv 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in ky around 0 61.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx} \cdot \frac{1}{\sin th}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{\left|\sin kx\right|}{ky \cdot \sin th}}\\ \mathbf{if}\;\sin th \leq -0.54:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\sin th \leq 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.77:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ (fabs (sin kx)) (* ky (sin th))))))
   (if (<= (sin th) -0.54)
     t_1
     (if (<= (sin th) 1e-9)
       (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
       (if (<= (sin th) 0.77)
         t_1
         (* (sin ky) (fabs (/ (sin th) (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 / (fabs(sin(kx)) / (ky * sin(th)));
	double tmp;
	if (sin(th) <= -0.54) {
		tmp = t_1;
	} else if (sin(th) <= 1e-9) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (sin(th) <= 0.77) {
		tmp = t_1;
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = 1.0 / (Math.abs(Math.sin(kx)) / (ky * Math.sin(th)));
	double tmp;
	if (Math.sin(th) <= -0.54) {
		tmp = t_1;
	} else if (Math.sin(th) <= 1e-9) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (Math.sin(th) <= 0.77) {
		tmp = t_1;
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = 1.0 / (math.fabs(math.sin(kx)) / (ky * math.sin(th)))
	tmp = 0
	if math.sin(th) <= -0.54:
		tmp = t_1
	elif math.sin(th) <= 1e-9:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif math.sin(th) <= 0.77:
		tmp = t_1
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 / Float64(abs(sin(kx)) / Float64(ky * sin(th))))
	tmp = 0.0
	if (sin(th) <= -0.54)
		tmp = t_1;
	elseif (sin(th) <= 1e-9)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (sin(th) <= 0.77)
		tmp = t_1;
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = 1.0 / (abs(sin(kx)) / (ky * sin(th)));
	tmp = 0.0;
	if (sin(th) <= -0.54)
		tmp = t_1;
	elseif (sin(th) <= 1e-9)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (sin(th) <= 0.77)
		tmp = t_1;
	else
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[th], $MachinePrecision], -0.54], t$95$1, If[LessEqual[N[Sin[th], $MachinePrecision], 1e-9], 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], 0.77], t$95$1, N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -0.54000000000000004 or 1.00000000000000006e-9 < (sin.f64 th) < 0.77000000000000002

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 93.8%

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

      7. associate-/l* 93.8%

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

      8. unpow2 93.8%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. clear-num 99.4%

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

      3. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ky around 0 20.6%

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

      1. add-sqr-sqrt 18.6%

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

      2. sqrt-prod 36.7%

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

      3. rem-sqrt-square 41.3%

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

   9. Applied egg-rr 41.3%

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

   if -0.54000000000000004 < (sin.f64 th) < 1.00000000000000006e-9

   1. Initial program 93.2%

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

      1. unpow2 93.2%

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

      2. sqr-neg 93.2%

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

      3. sin-neg 93.2%

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

      4. sin-neg 93.2%

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

      5. unpow2 93.2%

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

      6. associate-*l/ 90.4%

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

      7. associate-/l* 93.1%

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

      8. unpow2 93.1%

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

   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/ 92.9%

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

      2. clear-num 91.5%

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

      3. *-commutative 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in th around 0 79.9%

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

      1. *-commutative 79.9%

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

   9. Simplified 79.9%

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

       1. *-un-lft-identity 79.9%

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

       2. associate-/r* 86.2%

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

       3. associate-/r/ 88.0%

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

       4. clear-num 88.1%

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

   11. Applied egg-rr 88.1%

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

       1. *-lft-identity 88.1%

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

       2. associate-*l/ 81.3%

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

       3. associate-*r/ 88.0%

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

   13. Simplified 88.0%

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

   if 0.77000000000000002 < (sin.f64 th)

   1. Initial program 93.3%

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

      1. unpow2 93.3%

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

      2. sqr-neg 93.3%

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

      3. sin-neg 93.3%

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

      4. sin-neg 93.3%

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

      5. unpow2 93.3%

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

      6. associate-*l/ 93.3%

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

      7. associate-/l* 93.3%

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

      8. unpow2 93.3%

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

   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 20.9%

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

      1. add-sqr-sqrt 19.9%

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

      2. sqrt-unprod 37.5%

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

      3. pow2 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. unpow2 37.5%

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

      2. rem-sqrt-square 41.3%

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

   9. Simplified 41.3%

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

Alternative 8: 47.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.16:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-30} \lor \neg \left(\sin ky \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.16)
   (- th)
   (if (<= (sin ky) 4e-93)
     (* (sin th) (/ (sin ky) (sin kx)))
     (if (or (<= (sin ky) 5e-30) (not (<= (sin ky) 2e-29)))
       (sin th)
       (* ky (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.16) {
		tmp = -th;
	} else if (sin(ky) <= 4e-93) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if ((sin(ky) <= 5e-30) || !(sin(ky) <= 2e-29)) {
		tmp = sin(th);
	} else {
		tmp = ky * (sin(th) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.16d0)) then
        tmp = -th
    else if (sin(ky) <= 4d-93) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else if ((sin(ky) <= 5d-30) .or. (.not. (sin(ky) <= 2d-29))) then
        tmp = sin(th)
    else
        tmp = ky * (sin(th) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.16) {
		tmp = -th;
	} else if (Math.sin(ky) <= 4e-93) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else if ((Math.sin(ky) <= 5e-30) || !(Math.sin(ky) <= 2e-29)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.16:
		tmp = -th
	elif math.sin(ky) <= 4e-93:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	elif (math.sin(ky) <= 5e-30) or not (math.sin(ky) <= 2e-29):
		tmp = math.sin(th)
	else:
		tmp = ky * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.16)
		tmp = Float64(-th);
	elseif (sin(ky) <= 4e-93)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif ((sin(ky) <= 5e-30) || !(sin(ky) <= 2e-29))
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.16)
		tmp = -th;
	elseif (sin(ky) <= 4e-93)
		tmp = sin(th) * (sin(ky) / sin(kx));
	elseif ((sin(ky) <= 5e-30) || ~((sin(ky) <= 2e-29)))
		tmp = sin(th);
	else
		tmp = ky * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.16], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-93], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 5e-30], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 2e-29]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.160000000000000003

   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/ 99.5%

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

      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 99.5%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 3.4%

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

   7. Applied egg-rr 3.4%

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

      1. expm1-define 2.6%

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

      2. associate-*r/ 2.6%

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

      3. associate-*l/ 2.6%

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

      4. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 3.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 3.4%

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

       3. rem-exp-log 3.4%

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

       4. +-commutative 3.4%

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

   11. Applied egg-rr 3.4%

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

   12. Taylor expanded in th around 0 4.3%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 3.2%

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

       2. expm1-define 3.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 2.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 1.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 1.6%

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

       6. sqrt-unprod 10.4%

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

       7. pow2 10.4%

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

   14. Applied egg-rr 10.4%

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

   15. Taylor expanded in th around -inf 35.6%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 35.6%

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

   17. Simplified 35.6%

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

   if -0.160000000000000003 < (sin.f64 ky) < 3.9999999999999996e-93

   1. Initial program 84.0%

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

      1. +-commutative 84.0%

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

      2. unpow2 84.0%

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

      3. unpow2 84.0%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0 50.8%

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

   if 3.9999999999999996e-93 < (sin.f64 ky) < 4.99999999999999972e-30 or 1.99999999999999989e-29 < (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* 99.7%

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

      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 58.1%

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

   if 4.99999999999999972e-30 < (sin.f64 ky) < 1.99999999999999989e-29

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 91.6%

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

      7. associate-/l* 93.3%

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

      8. unpow2 93.3%

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

   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 20.4%

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

      1. associate-/l* 22.2%

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

   7. Simplified 22.2%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.16:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-30} \lor \neg \left(\sin ky \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -0.16:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\sin ky \cdot t\_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-30} \lor \neg \left(\sin ky \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin ky) -0.16)
     (- th)
     (if (<= (sin ky) 4e-93)
       (* (sin ky) t_1)
       (if (or (<= (sin ky) 5e-30) (not (<= (sin ky) 2e-29)))
         (sin th)
         (* ky t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(ky) <= -0.16) {
		tmp = -th;
	} else if (sin(ky) <= 4e-93) {
		tmp = sin(ky) * t_1;
	} else if ((sin(ky) <= 5e-30) || !(sin(ky) <= 2e-29)) {
		tmp = sin(th);
	} else {
		tmp = ky * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    if (sin(ky) <= (-0.16d0)) then
        tmp = -th
    else if (sin(ky) <= 4d-93) then
        tmp = sin(ky) * t_1
    else if ((sin(ky) <= 5d-30) .or. (.not. (sin(ky) <= 2d-29))) then
        tmp = sin(th)
    else
        tmp = ky * t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(ky) <= -0.16) {
		tmp = -th;
	} else if (Math.sin(ky) <= 4e-93) {
		tmp = Math.sin(ky) * t_1;
	} else if ((Math.sin(ky) <= 5e-30) || !(Math.sin(ky) <= 2e-29)) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(ky) <= -0.16:
		tmp = -th
	elif math.sin(ky) <= 4e-93:
		tmp = math.sin(ky) * t_1
	elif (math.sin(ky) <= 5e-30) or not (math.sin(ky) <= 2e-29):
		tmp = math.sin(th)
	else:
		tmp = ky * t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(ky) <= -0.16)
		tmp = Float64(-th);
	elseif (sin(ky) <= 4e-93)
		tmp = Float64(sin(ky) * t_1);
	elseif ((sin(ky) <= 5e-30) || !(sin(ky) <= 2e-29))
		tmp = sin(th);
	else
		tmp = Float64(ky * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(ky) <= -0.16)
		tmp = -th;
	elseif (sin(ky) <= 4e-93)
		tmp = sin(ky) * t_1;
	elseif ((sin(ky) <= 5e-30) || ~((sin(ky) <= 2e-29)))
		tmp = sin(th);
	else
		tmp = ky * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.16], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-93], N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 5e-30], N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 2e-29]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(ky * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.160000000000000003

   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/ 99.5%

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

      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 99.5%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 3.4%

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

   7. Applied egg-rr 3.4%

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

      1. expm1-define 2.6%

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

      2. associate-*r/ 2.6%

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

      3. associate-*l/ 2.6%

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

      4. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 3.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 3.4%

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

       3. rem-exp-log 3.4%

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

       4. +-commutative 3.4%

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

   11. Applied egg-rr 3.4%

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

   12. Taylor expanded in th around 0 4.3%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 3.2%

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

       2. expm1-define 3.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 2.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 1.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 1.6%

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

       6. sqrt-unprod 10.4%

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

       7. pow2 10.4%

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

   14. Applied egg-rr 10.4%

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

   15. Taylor expanded in th around -inf 35.6%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 35.6%

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

   17. Simplified 35.6%

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

   if -0.160000000000000003 < (sin.f64 ky) < 3.9999999999999996e-93

   1. Initial program 84.0%

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

      1. unpow2 84.0%

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

      2. sqr-neg 84.0%

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

      3. sin-neg 84.0%

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

      4. sin-neg 84.0%

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

      5. unpow2 84.0%

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

      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* 83.9%

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

      8. unpow2 83.9%

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

   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 50.8%

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

   if 3.9999999999999996e-93 < (sin.f64 ky) < 4.99999999999999972e-30 or 1.99999999999999989e-29 < (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* 99.7%

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

      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 58.1%

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

   if 4.99999999999999972e-30 < (sin.f64 ky) < 1.99999999999999989e-29

   1. Initial program 93.4%

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

      1. unpow2 93.4%

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

      2. sqr-neg 93.4%

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

      3. sin-neg 93.4%

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

      4. sin-neg 93.4%

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

      5. unpow2 93.4%

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

      6. associate-*l/ 91.6%

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

      7. associate-/l* 93.3%

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

      8. unpow2 93.3%

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

   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 20.4%

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

      1. associate-/l* 22.2%

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

   7. Simplified 22.2%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.16:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-30} \lor \neg \left(\sin ky \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. unpow2 93.4%

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

   2. sqr-neg 93.4%

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

   3. sin-neg 93.4%

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

   4. sin-neg 93.4%

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

   5. unpow2 93.4%

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

   6. associate-*l/ 91.6%

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

   7. associate-/l* 93.3%

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

   8. unpow2 93.3%

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

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. Add Preprocessing

Alternative 11: 63.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq 0.00033:\\ \;\;\;\;\frac{\sin th}{t\_1 \cdot \frac{1}{ky}}\\ \mathbf{elif}\;ky \leq 1.02 \cdot 10^{+111}:\\ \;\;\;\;\sin ky \cdot \frac{th}{t\_1}\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{+188}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= ky 0.00033)
     (/ (sin th) (* t_1 (/ 1.0 ky)))
     (if (<= ky 1.02e+111)
       (* (sin ky) (/ th t_1))
       (if (<= ky 2.7e+188)
         (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
         (* (/ (sin ky) t_1) th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (ky <= 0.00033) {
		tmp = sin(th) / (t_1 * (1.0 / ky));
	} else if (ky <= 1.02e+111) {
		tmp = sin(ky) * (th / t_1);
	} else if (ky <= 2.7e+188) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = (sin(ky) / t_1) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (ky <= 0.00033) {
		tmp = Math.sin(th) / (t_1 * (1.0 / ky));
	} else if (ky <= 1.02e+111) {
		tmp = Math.sin(ky) * (th / t_1);
	} else if (ky <= 2.7e+188) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = (Math.sin(ky) / t_1) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if ky <= 0.00033:
		tmp = math.sin(th) / (t_1 * (1.0 / ky))
	elif ky <= 1.02e+111:
		tmp = math.sin(ky) * (th / t_1)
	elif ky <= 2.7e+188:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = (math.sin(ky) / t_1) * th
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (ky <= 0.00033)
		tmp = Float64(sin(th) / Float64(t_1 * Float64(1.0 / ky)));
	elseif (ky <= 1.02e+111)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	elseif (ky <= 2.7e+188)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (ky <= 0.00033)
		tmp = sin(th) / (t_1 * (1.0 / ky));
	elseif (ky <= 1.02e+111)
		tmp = sin(ky) * (th / t_1);
	elseif (ky <= 2.7e+188)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = (sin(ky) / t_1) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[ky, 0.00033], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 1.02e+111], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.7e+188], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < 3.3e-4

   1. Initial program 91.1%

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

      1. +-commutative 91.1%

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

      2. unpow2 91.1%

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

      3. unpow2 91.1%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-inv 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in ky around 0 59.5%

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

   if 3.3e-4 < ky < 1.02e111

   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/ 99.6%

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

      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. 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. clear-num 97.2%

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

      3. *-commutative 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in th around 0 65.0%

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

      1. *-commutative 65.0%

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

   9. Simplified 65.0%

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

       1. *-un-lft-identity 65.0%

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

       2. associate-/r* 65.0%

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

       3. associate-/r/ 67.6%

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

       4. clear-num 67.6%

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

   11. Applied egg-rr 67.6%

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

       1. *-lft-identity 67.6%

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

       2. associate-*l/ 67.5%

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

       3. associate-*r/ 67.5%

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

   13. Simplified 67.5%

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

   if 1.02e111 < ky < 2.7e188

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in kx around 0 45.1%

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

   if 2.7e188 < 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/ 99.5%

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

      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. 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. clear-num 99.2%

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

      3. *-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in th around 0 65.2%

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

      1. *-commutative 65.2%

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

   9. Simplified 65.2%

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

       1. associate-/r* 65.2%

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

       2. associate-/r/ 65.3%

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

       3. clear-num 65.4%

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

   11. Applied egg-rr 65.4%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.00033:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{ky}}\\ \mathbf{elif}\;ky \leq 1.02 \cdot 10^{+111}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{+188}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{1}{\frac{\left|\sin kx\right|}{ky \cdot \sin th}}\\ \mathbf{elif}\;ky \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin ky}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.2e-84)
   (/ 1.0 (/ (fabs (sin kx)) (* ky (sin th))))
   (if (<= ky 1.6e-35) (sin th) (* (sin ky) (fabs (/ (sin th) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.2e-84) {
		tmp = 1.0 / (fabs(sin(kx)) / (ky * sin(th)));
	} else if (ky <= 1.6e-35) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(ky)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.2d-84) then
        tmp = 1.0d0 / (abs(sin(kx)) / (ky * sin(th)))
    else if (ky <= 1.6d-35) then
        tmp = sin(th)
    else
        tmp = sin(ky) * abs((sin(th) / sin(ky)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.2e-84) {
		tmp = 1.0 / (Math.abs(Math.sin(kx)) / (ky * Math.sin(th)));
	} else if (ky <= 1.6e-35) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.2e-84:
		tmp = 1.0 / (math.fabs(math.sin(kx)) / (ky * math.sin(th)))
	elif ky <= 1.6e-35:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.2e-84)
		tmp = Float64(1.0 / Float64(abs(sin(kx)) / Float64(ky * sin(th))));
	elseif (ky <= 1.6e-35)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.2e-84)
		tmp = 1.0 / (abs(sin(kx)) / (ky * sin(th)));
	elseif (ky <= 1.6e-35)
		tmp = sin(th);
	else
		tmp = sin(ky) * abs((sin(th) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.2e-84], N[(1.0 / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 1.6e-35], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.20000000000000009e-84

   1. Initial program 90.3%

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

      1. unpow2 90.3%

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

      2. sqr-neg 90.3%

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

      3. sin-neg 90.3%

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

      4. sin-neg 90.3%

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

      5. unpow2 90.3%

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

      6. associate-*l/ 88.3%

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

      7. associate-/l* 90.3%

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

      8. unpow2 90.3%

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

   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/ 94.0%

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

      2. clear-num 93.1%

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

      3. *-commutative 93.1%

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

   6. Applied egg-rr 93.1%

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

   7. Taylor expanded in ky around 0 28.1%

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

      1. add-sqr-sqrt 21.3%

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

      2. sqrt-prod 39.4%

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

      3. rem-sqrt-square 43.6%

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

   9. Applied egg-rr 43.6%

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

   if 1.20000000000000009e-84 < ky < 1.5999999999999999e-35

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 89.7%

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

      7. associate-/l* 99.8%

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

      8. unpow2 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 79.3%

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

   if 1.5999999999999999e-35 < 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/ 99.5%

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

      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 35.3%

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

      1. add-sqr-sqrt 21.0%

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

      2. sqrt-unprod 31.1%

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

      3. pow2 31.1%

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

   7. Applied egg-rr 31.1%

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

      1. unpow2 31.1%

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

      2. rem-sqrt-square 40.4%

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

   9. Simplified 40.4%

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

Alternative 13: 47.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.116:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.116)
   (- th)
   (if (<= (sin ky) 4e-93) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.116) {
		tmp = -th;
	} else if (sin(ky) <= 4e-93) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.116d0)) then
        tmp = -th
    else if (sin(ky) <= 4d-93) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.116) {
		tmp = -th;
	} else if (Math.sin(ky) <= 4e-93) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.116:
		tmp = -th
	elif math.sin(ky) <= 4e-93:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.116)
		tmp = Float64(-th);
	elseif (sin(ky) <= 4e-93)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.116)
		tmp = -th;
	elseif (sin(ky) <= 4e-93)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.116], (-th), If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-93], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.116000000000000006

   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/ 99.5%

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

      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 99.5%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 3.4%

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

   7. Applied egg-rr 3.4%

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

      1. expm1-define 2.6%

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

      2. associate-*r/ 2.6%

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

      3. associate-*l/ 2.6%

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

      4. *-inverses 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 2.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 3.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 3.4%

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

       3. rem-exp-log 3.4%

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

       4. +-commutative 3.4%

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

   11. Applied egg-rr 3.4%

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

   12. Taylor expanded in th around 0 4.3%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 3.2%

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

       2. expm1-define 3.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 2.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 1.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 1.6%

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

       6. sqrt-unprod 10.1%

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

       7. pow2 10.1%

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

   14. Applied egg-rr 10.1%

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

   15. Taylor expanded in th around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 36.1%

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

   17. Simplified 36.1%

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

   if -0.116000000000000006 < (sin.f64 ky) < 3.9999999999999996e-93

   1. Initial program 83.7%

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

   3. Taylor expanded in ky around 0 50.8%

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

   if 3.9999999999999996e-93 < (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* 99.7%

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

      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 58.1%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.116:\\ \;\;\;\;-th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 135000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;th \leq 3.8 \cdot 10^{+161}:\\ \;\;\;\;\frac{1}{\frac{\left|\sin kx\right|}{ky \cdot \sin th}}\\ \mathbf{elif}\;th \leq 2.7 \cdot 10^{+218}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 135000.0)
   (* th (/ (sin ky) (hypot (sin ky) kx)))
   (if (<= th 3.8e+161)
     (/ 1.0 (/ (fabs (sin kx)) (* ky (sin th))))
     (if (<= th 2.7e+218) (sin th) (* (sin ky) (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 135000.0) {
		tmp = th * (sin(ky) / hypot(sin(ky), kx));
	} else if (th <= 3.8e+161) {
		tmp = 1.0 / (fabs(sin(kx)) / (ky * sin(th)));
	} else if (th <= 2.7e+218) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 135000.0) {
		tmp = th * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else if (th <= 3.8e+161) {
		tmp = 1.0 / (Math.abs(Math.sin(kx)) / (ky * Math.sin(th)));
	} else if (th <= 2.7e+218) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 135000.0:
		tmp = th * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	elif th <= 3.8e+161:
		tmp = 1.0 / (math.fabs(math.sin(kx)) / (ky * math.sin(th)))
	elif th <= 2.7e+218:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 135000.0)
		tmp = Float64(th * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (th <= 3.8e+161)
		tmp = Float64(1.0 / Float64(abs(sin(kx)) / Float64(ky * sin(th))));
	elseif (th <= 2.7e+218)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 135000.0)
		tmp = th * (sin(ky) / hypot(sin(ky), kx));
	elseif (th <= 3.8e+161)
		tmp = 1.0 / (abs(sin(kx)) / (ky * sin(th)));
	elseif (th <= 2.7e+218)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 135000.0], N[(th * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.8e+161], N[(1.0 / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 2.7e+218], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if th < 135000

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 91.5%

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

      7. associate-/l* 93.8%

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

      8. unpow2 93.8%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 94.1%

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

      2. clear-num 92.9%

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

      3. *-commutative 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in th around 0 66.5%

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

      1. *-commutative 66.5%

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

   9. Simplified 66.5%

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

       1. associate-/r* 71.7%

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

       2. associate-/r/ 73.2%

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

       3. clear-num 73.3%

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

   11. Applied egg-rr 73.3%

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

   12. Taylor expanded in kx around 0 46.9%

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

   if 135000 < th < 3.8000000000000002e161

   1. Initial program 93.3%

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

      1. unpow2 93.3%

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

      2. sqr-neg 93.3%

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

      3. sin-neg 93.3%

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

      4. sin-neg 93.3%

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

      5. unpow2 93.3%

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

      6. associate-*l/ 93.1%

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

      7. associate-/l* 93.0%

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

      8. unpow2 93.0%

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

   3. Simplified 99.4%

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

      1. associate-*r/ 99.5%

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

      2. clear-num 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in ky around 0 18.7%

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

      1. add-sqr-sqrt 17.3%

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

      2. sqrt-prod 43.8%

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

      3. rem-sqrt-square 50.4%

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

   9. Applied egg-rr 50.4%

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

   if 3.8000000000000002e161 < th < 2.70000000000000013e218

   1. Initial program 92.2%

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

      1. unpow2 92.2%

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

      2. sqr-neg 92.2%

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

      3. sin-neg 92.2%

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

      4. sin-neg 92.2%

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

      5. unpow2 92.2%

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

      6. associate-*l/ 92.1%

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

      7. associate-/l* 92.2%

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

      8. unpow2 92.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 23.4%

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

   if 2.70000000000000013e218 < th

   1. Initial program 90.0%

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

      1. unpow2 90.0%

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

      2. sqr-neg 90.0%

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

      3. sin-neg 90.0%

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

      4. sin-neg 90.0%

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

      5. unpow2 90.0%

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

      6. associate-*l/ 90.0%

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

      7. associate-/l* 90.2%

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

      8. unpow2 90.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ky around 0 33.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 135000:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;th \leq 3.8 \cdot 10^{+161}:\\ \;\;\;\;\frac{1}{\frac{\left|\sin kx\right|}{ky \cdot \sin th}}\\ \mathbf{elif}\;th \leq 2.7 \cdot 10^{+218}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin th\right|\\ t_2 := ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{if}\;ky \leq 2 \cdot 10^{-85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ky \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ky \leq 9 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ky \leq 6.1 \cdot 10^{+134}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{+205}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 4.4 \cdot 10^{+223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ky \leq 9.8 \cdot 10^{+275}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin th))) (t_2 (* ky (/ (sin th) (sin kx)))))
   (if (<= ky 2e-85)
     t_2
     (if (<= ky 4.9e-30)
       (sin th)
       (if (<= ky 1.75e-8)
         t_2
         (if (<= ky 9e+98)
           t_1
           (if (<= ky 6.1e+134)
             (sin th)
             (if (<= ky 4.8e+166)
               t_1
               (if (<= ky 2.7e+205)
                 (- th)
                 (if (<= ky 4.4e+223)
                   t_1
                   (if (<= ky 9.8e+275) (- th) (sin th))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(th));
	double t_2 = ky * (sin(th) / sin(kx));
	double tmp;
	if (ky <= 2e-85) {
		tmp = t_2;
	} else if (ky <= 4.9e-30) {
		tmp = sin(th);
	} else if (ky <= 1.75e-8) {
		tmp = t_2;
	} else if (ky <= 9e+98) {
		tmp = t_1;
	} else if (ky <= 6.1e+134) {
		tmp = sin(th);
	} else if (ky <= 4.8e+166) {
		tmp = t_1;
	} else if (ky <= 2.7e+205) {
		tmp = -th;
	} else if (ky <= 4.4e+223) {
		tmp = t_1;
	} else if (ky <= 9.8e+275) {
		tmp = -th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs(sin(th))
    t_2 = ky * (sin(th) / sin(kx))
    if (ky <= 2d-85) then
        tmp = t_2
    else if (ky <= 4.9d-30) then
        tmp = sin(th)
    else if (ky <= 1.75d-8) then
        tmp = t_2
    else if (ky <= 9d+98) then
        tmp = t_1
    else if (ky <= 6.1d+134) then
        tmp = sin(th)
    else if (ky <= 4.8d+166) then
        tmp = t_1
    else if (ky <= 2.7d+205) then
        tmp = -th
    else if (ky <= 4.4d+223) then
        tmp = t_1
    else if (ky <= 9.8d+275) then
        tmp = -th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(th));
	double t_2 = ky * (Math.sin(th) / Math.sin(kx));
	double tmp;
	if (ky <= 2e-85) {
		tmp = t_2;
	} else if (ky <= 4.9e-30) {
		tmp = Math.sin(th);
	} else if (ky <= 1.75e-8) {
		tmp = t_2;
	} else if (ky <= 9e+98) {
		tmp = t_1;
	} else if (ky <= 6.1e+134) {
		tmp = Math.sin(th);
	} else if (ky <= 4.8e+166) {
		tmp = t_1;
	} else if (ky <= 2.7e+205) {
		tmp = -th;
	} else if (ky <= 4.4e+223) {
		tmp = t_1;
	} else if (ky <= 9.8e+275) {
		tmp = -th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(th))
	t_2 = ky * (math.sin(th) / math.sin(kx))
	tmp = 0
	if ky <= 2e-85:
		tmp = t_2
	elif ky <= 4.9e-30:
		tmp = math.sin(th)
	elif ky <= 1.75e-8:
		tmp = t_2
	elif ky <= 9e+98:
		tmp = t_1
	elif ky <= 6.1e+134:
		tmp = math.sin(th)
	elif ky <= 4.8e+166:
		tmp = t_1
	elif ky <= 2.7e+205:
		tmp = -th
	elif ky <= 4.4e+223:
		tmp = t_1
	elif ky <= 9.8e+275:
		tmp = -th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(th))
	t_2 = Float64(ky * Float64(sin(th) / sin(kx)))
	tmp = 0.0
	if (ky <= 2e-85)
		tmp = t_2;
	elseif (ky <= 4.9e-30)
		tmp = sin(th);
	elseif (ky <= 1.75e-8)
		tmp = t_2;
	elseif (ky <= 9e+98)
		tmp = t_1;
	elseif (ky <= 6.1e+134)
		tmp = sin(th);
	elseif (ky <= 4.8e+166)
		tmp = t_1;
	elseif (ky <= 2.7e+205)
		tmp = Float64(-th);
	elseif (ky <= 4.4e+223)
		tmp = t_1;
	elseif (ky <= 9.8e+275)
		tmp = Float64(-th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(th));
	t_2 = ky * (sin(th) / sin(kx));
	tmp = 0.0;
	if (ky <= 2e-85)
		tmp = t_2;
	elseif (ky <= 4.9e-30)
		tmp = sin(th);
	elseif (ky <= 1.75e-8)
		tmp = t_2;
	elseif (ky <= 9e+98)
		tmp = t_1;
	elseif (ky <= 6.1e+134)
		tmp = sin(th);
	elseif (ky <= 4.8e+166)
		tmp = t_1;
	elseif (ky <= 2.7e+205)
		tmp = -th;
	elseif (ky <= 4.4e+223)
		tmp = t_1;
	elseif (ky <= 9.8e+275)
		tmp = -th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ky, 2e-85], t$95$2, If[LessEqual[ky, 4.9e-30], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 1.75e-8], t$95$2, If[LessEqual[ky, 9e+98], t$95$1, If[LessEqual[ky, 6.1e+134], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 4.8e+166], t$95$1, If[LessEqual[ky, 2.7e+205], (-th), If[LessEqual[ky, 4.4e+223], t$95$1, If[LessEqual[ky, 9.8e+275], (-th), N[Sin[th], $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if ky < 2e-85 or 4.89999999999999971e-30 < ky < 1.75000000000000012e-8

   1. Initial program 90.5%

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

      1. unpow2 90.5%

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

      2. sqr-neg 90.5%

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

      3. sin-neg 90.5%

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

      4. sin-neg 90.5%

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

      5. unpow2 90.5%

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

      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* 90.4%

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

      8. unpow2 90.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 28.3%

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

      1. associate-/l* 31.0%

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

   7. Simplified 31.0%

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

   if 2e-85 < ky < 4.89999999999999971e-30 or 9.0000000000000004e98 < ky < 6.09999999999999978e134 or 9.7999999999999996e275 < ky

   1. Initial program 99.9%

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

      1. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. sin-neg 99.9%

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

      4. sin-neg 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*l/ 96.8%

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

      7. associate-/l* 99.8%

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

      8. unpow2 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 57.3%

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

   if 1.75000000000000012e-8 < ky < 9.0000000000000004e98 or 6.09999999999999978e134 < ky < 4.79999999999999984e166 or 2.70000000000000012e205 < ky < 4.3999999999999999e223

   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/ 99.5%

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

      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 33.3%

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

      1. add-sqr-sqrt 18.2%

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

      2. sqrt-unprod 22.4%

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

      3. pow2 22.4%

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

   7. Applied egg-rr 22.4%

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

      1. unpow2 22.4%

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

      2. rem-sqrt-square 27.8%

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

      3. associate-*r/ 27.7%

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

      4. associate-*l/ 27.8%

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

      5. *-inverses 27.8%

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

      6. *-lft-identity 27.8%

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

   9. Simplified 27.8%

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

   if 4.79999999999999984e166 < ky < 2.70000000000000012e205 or 4.3999999999999999e223 < ky < 9.7999999999999996e275

   1. Initial program 99.4%

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

      1. unpow2 99.4%

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

      2. sqr-neg 99.4%

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

      3. sin-neg 99.4%

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

      4. sin-neg 99.4%

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

      5. unpow2 99.4%

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

      6. associate-*l/ 99.3%

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

      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 13.8%

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

      1. expm1-log1p-u 13.7%

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

      2. expm1-undefine 12.2%

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

   7. Applied egg-rr 12.2%

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

      1. expm1-define 13.7%

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

      2. associate-*r/ 13.7%

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

      3. associate-*l/ 13.7%

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

      4. *-inverses 13.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 13.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 13.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 12.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 12.2%

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

       3. rem-exp-log 12.2%

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

       4. +-commutative 12.2%

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

   11. Applied egg-rr 12.2%

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

   12. Taylor expanded in th around 0 4.9%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 4.4%

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

       2. expm1-define 4.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 5.8%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 3.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 3.8%

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

       6. sqrt-unprod 11.5%

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

       7. pow2 11.5%

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

   14. Applied egg-rr 11.5%

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

   15. Taylor expanded in th around -inf 39.7%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 39.7%

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

   17. Simplified 39.7%

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

Alternative 16: 27.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7 \cdot 10^{-153}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 4.8 \cdot 10^{+201}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 7e-153)
   (* ky (/ (sin th) kx))
   (if (or (<= ky 8.2e+158) (not (<= ky 4.8e+201))) (sin th) (- th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7e-153) {
		tmp = ky * (sin(th) / kx);
	} else if ((ky <= 8.2e+158) || !(ky <= 4.8e+201)) {
		tmp = sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 7d-153) then
        tmp = ky * (sin(th) / kx)
    else if ((ky <= 8.2d+158) .or. (.not. (ky <= 4.8d+201))) then
        tmp = sin(th)
    else
        tmp = -th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 7e-153) {
		tmp = ky * (Math.sin(th) / kx);
	} else if ((ky <= 8.2e+158) || !(ky <= 4.8e+201)) {
		tmp = Math.sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 7e-153:
		tmp = ky * (math.sin(th) / kx)
	elif (ky <= 8.2e+158) or not (ky <= 4.8e+201):
		tmp = math.sin(th)
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 7e-153)
		tmp = Float64(ky * Float64(sin(th) / kx));
	elseif ((ky <= 8.2e+158) || !(ky <= 4.8e+201))
		tmp = sin(th);
	else
		tmp = Float64(-th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 7e-153)
		tmp = ky * (sin(th) / kx);
	elseif ((ky <= 8.2e+158) || ~((ky <= 4.8e+201)))
		tmp = sin(th);
	else
		tmp = -th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 7e-153], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[ky, 8.2e+158], N[Not[LessEqual[ky, 4.8e+201]], $MachinePrecision]], N[Sin[th], $MachinePrecision], (-th)]]

Derivation

1. Split input into 3 regimes

2. if ky < 6.99999999999999961e-153

   1. Initial program 90.1%

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

      1. unpow2 90.1%

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

      2. sqr-neg 90.1%

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

      3. sin-neg 90.1%

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

      4. sin-neg 90.1%

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

      5. unpow2 90.1%

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

      6. associate-*l/ 88.0%

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

      7. associate-/l* 90.0%

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

      8. unpow2 90.0%

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

   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. associate-*r/ 93.8%

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

      2. clear-num 92.9%

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

      3. *-commutative 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in ky around 0 26.5%

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

   8. Taylor expanded in kx around 0 18.2%

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

      1. associate-/l* 21.0%

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

   10. Simplified 21.0%

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

   if 6.99999999999999961e-153 < ky < 8.20000000000000008e158 or 4.79999999999999985e201 < 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 40.3%

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

   if 8.20000000000000008e158 < ky < 4.79999999999999985e201

   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/ 99.2%

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

      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 24.2%

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

      1. expm1-log1p-u 24.2%

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

      2. expm1-undefine 24.2%

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

   7. Applied egg-rr 24.2%

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

      1. expm1-define 24.2%

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

      2. associate-*r/ 23.8%

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

      3. associate-*l/ 24.2%

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

      4. *-inverses 24.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 24.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 24.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 24.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 24.2%

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

       3. rem-exp-log 24.2%

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

       4. +-commutative 24.2%

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

   11. Applied egg-rr 24.2%

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

   12. Taylor expanded in th around 0 6.7%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 5.7%

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

       2. expm1-define 5.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 5.7%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 5.1%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 5.1%

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

       6. sqrt-unprod 5.5%

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

       7. pow2 5.5%

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

   14. Applied egg-rr 5.5%

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

   15. Taylor expanded in th around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 36.1%

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

   17. Simplified 36.1%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 7 \cdot 10^{-153}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 4.8 \cdot 10^{+201}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{-160}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 2.7 \cdot 10^{+205}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 6e-160)
   (* ky (/ th (sin kx)))
   (if (or (<= ky 8.2e+158) (not (<= ky 2.7e+205))) (sin th) (- th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6e-160) {
		tmp = ky * (th / sin(kx));
	} else if ((ky <= 8.2e+158) || !(ky <= 2.7e+205)) {
		tmp = sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 6d-160) then
        tmp = ky * (th / sin(kx))
    else if ((ky <= 8.2d+158) .or. (.not. (ky <= 2.7d+205))) then
        tmp = sin(th)
    else
        tmp = -th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 6e-160) {
		tmp = ky * (th / Math.sin(kx));
	} else if ((ky <= 8.2e+158) || !(ky <= 2.7e+205)) {
		tmp = Math.sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 6e-160:
		tmp = ky * (th / math.sin(kx))
	elif (ky <= 8.2e+158) or not (ky <= 2.7e+205):
		tmp = math.sin(th)
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 6e-160)
		tmp = Float64(ky * Float64(th / sin(kx)));
	elseif ((ky <= 8.2e+158) || !(ky <= 2.7e+205))
		tmp = sin(th);
	else
		tmp = Float64(-th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 6e-160)
		tmp = ky * (th / sin(kx));
	elseif ((ky <= 8.2e+158) || ~((ky <= 2.7e+205)))
		tmp = sin(th);
	else
		tmp = -th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 6e-160], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[ky, 8.2e+158], N[Not[LessEqual[ky, 2.7e+205]], $MachinePrecision]], N[Sin[th], $MachinePrecision], (-th)]]

Derivation

1. Split input into 3 regimes

2. if ky < 5.99999999999999993e-160

   1. Initial program 90.1%

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

      1. unpow2 90.1%

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

      2. sqr-neg 90.1%

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

      3. sin-neg 90.1%

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

      4. sin-neg 90.1%

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

      5. unpow2 90.1%

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

      6. associate-*l/ 88.0%

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

      7. associate-/l* 90.0%

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

      8. unpow2 90.0%

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

   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. associate-*r/ 93.8%

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

      2. clear-num 92.9%

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

      3. *-commutative 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in ky around 0 26.5%

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

   8. Taylor expanded in th around 0 16.0%

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

      1. associate-/l* 18.8%

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

   10. Simplified 18.8%

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

   if 5.99999999999999993e-160 < ky < 8.20000000000000008e158 or 2.70000000000000012e205 < 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 40.7%

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

   if 8.20000000000000008e158 < ky < 2.70000000000000012e205

   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/ 99.3%

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

      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 21.6%

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

      1. expm1-log1p-u 21.6%

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

      2. expm1-undefine 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. expm1-define 21.6%

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

      2. associate-*r/ 21.2%

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

      3. associate-*l/ 21.6%

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

      4. *-inverses 21.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 21.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 21.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 21.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 21.8%

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

       3. rem-exp-log 21.8%

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

       4. +-commutative 21.8%

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

   11. Applied egg-rr 21.8%

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

   12. Taylor expanded in th around 0 6.8%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 5.9%

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

       2. expm1-define 5.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 5.7%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 5.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 5.2%

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

       6. sqrt-unprod 5.8%

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

       7. pow2 5.8%

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

   14. Applied egg-rr 5.8%

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

   15. Taylor expanded in th around -inf 45.2%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 45.2%

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

   17. Simplified 45.2%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{-160}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 2.7 \cdot 10^{+205}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 23.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.25 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{log1p}\left(0\right)\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 2.7 \cdot 10^{+205}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.25e-178)
   (log1p 0.0)
   (if (or (<= ky 8.2e+158) (not (<= ky 2.7e+205))) (sin th) (- th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.25e-178) {
		tmp = log1p(0.0);
	} else if ((ky <= 8.2e+158) || !(ky <= 2.7e+205)) {
		tmp = sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.25e-178) {
		tmp = Math.log1p(0.0);
	} else if ((ky <= 8.2e+158) || !(ky <= 2.7e+205)) {
		tmp = Math.sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.25e-178:
		tmp = math.log1p(0.0)
	elif (ky <= 8.2e+158) or not (ky <= 2.7e+205):
		tmp = math.sin(th)
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.25e-178)
		tmp = log1p(0.0);
	elseif ((ky <= 8.2e+158) || !(ky <= 2.7e+205))
		tmp = sin(th);
	else
		tmp = Float64(-th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.25e-178], N[Log[1 + 0.0], $MachinePrecision], If[Or[LessEqual[ky, 8.2e+158], N[Not[LessEqual[ky, 2.7e+205]], $MachinePrecision]], N[Sin[th], $MachinePrecision], (-th)]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.2500000000000001e-178

   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/ 89.5%

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

      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 99.6%

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

   5. Taylor expanded in kx around 0 17.4%

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

      1. expm1-log1p-u 17.4%

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

      2. expm1-undefine 18.8%

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

   7. Applied egg-rr 18.8%

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

      1. expm1-define 17.4%

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

      2. associate-*r/ 24.6%

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

      3. associate-*l/ 17.4%

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

      4. *-inverses 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 17.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 17.4%

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

       2. log1p-expm1-u 17.3%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin th\right)\right)} \]

   11. Applied egg-rr 17.3%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin th\right)\right)} \]

   12. Taylor expanded in th around inf 18.8%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin th} - 1}\right) \]

   13. Taylor expanded in th around 0 13.3%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{1} - 1\right) \]

   if 4.2500000000000001e-178 < ky < 8.20000000000000008e158 or 2.70000000000000012e205 < ky

   1. Initial program 96.9%

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

      1. unpow2 96.9%

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

      2. sqr-neg 96.9%

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

      3. sin-neg 96.9%

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

      4. sin-neg 96.9%

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

      5. unpow2 96.9%

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

      6. associate-*l/ 94.8%

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

      7. associate-/l* 96.9%

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

      8. unpow2 96.9%

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

   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 38.4%

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

   if 8.20000000000000008e158 < ky < 2.70000000000000012e205

   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/ 99.3%

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

      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 21.6%

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

      1. expm1-log1p-u 21.6%

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

      2. expm1-undefine 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. expm1-define 21.6%

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

      2. associate-*r/ 21.2%

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

      3. associate-*l/ 21.6%

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

      4. *-inverses 21.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 21.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 21.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 21.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 21.8%

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

       3. rem-exp-log 21.8%

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

       4. +-commutative 21.8%

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

   11. Applied egg-rr 21.8%

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

   12. Taylor expanded in th around 0 6.8%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 5.9%

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

       2. expm1-define 5.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 5.7%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 5.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 5.2%

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

       6. sqrt-unprod 5.8%

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

       7. pow2 5.8%

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

   14. Applied egg-rr 5.8%

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

   15. Taylor expanded in th around -inf 45.2%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 45.2%

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

   17. Simplified 45.2%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.25 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{log1p}\left(0\right)\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 2.7 \cdot 10^{+205}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 23.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 2.7 \cdot 10^{+205}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.15e-160)
   (+ (+ th 1.0) -1.0)
   (if (or (<= ky 8.2e+158) (not (<= ky 2.7e+205))) (sin th) (- th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.15e-160) {
		tmp = (th + 1.0) + -1.0;
	} else if ((ky <= 8.2e+158) || !(ky <= 2.7e+205)) {
		tmp = sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.15d-160) then
        tmp = (th + 1.0d0) + (-1.0d0)
    else if ((ky <= 8.2d+158) .or. (.not. (ky <= 2.7d+205))) then
        tmp = sin(th)
    else
        tmp = -th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.15e-160) {
		tmp = (th + 1.0) + -1.0;
	} else if ((ky <= 8.2e+158) || !(ky <= 2.7e+205)) {
		tmp = Math.sin(th);
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.15e-160:
		tmp = (th + 1.0) + -1.0
	elif (ky <= 8.2e+158) or not (ky <= 2.7e+205):
		tmp = math.sin(th)
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.15e-160)
		tmp = Float64(Float64(th + 1.0) + -1.0);
	elseif ((ky <= 8.2e+158) || !(ky <= 2.7e+205))
		tmp = sin(th);
	else
		tmp = Float64(-th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.15e-160)
		tmp = (th + 1.0) + -1.0;
	elseif ((ky <= 8.2e+158) || ~((ky <= 2.7e+205)))
		tmp = sin(th);
	else
		tmp = -th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.15e-160], N[(N[(th + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[ky, 8.2e+158], N[Not[LessEqual[ky, 2.7e+205]], $MachinePrecision]], N[Sin[th], $MachinePrecision], (-th)]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.14999999999999992e-160

   1. Initial program 90.1%

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

      1. unpow2 90.1%

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

      2. sqr-neg 90.1%

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

      3. sin-neg 90.1%

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

      4. sin-neg 90.1%

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

      5. unpow2 90.1%

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

      6. associate-*l/ 88.0%

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

      7. associate-/l* 90.0%

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

      8. unpow2 90.0%

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

   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. Taylor expanded in kx around 0 17.4%

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

      1. expm1-log1p-u 17.4%

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

      2. expm1-undefine 19.3%

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

   7. Applied egg-rr 19.3%

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

      1. expm1-define 17.4%

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

      2. associate-*r/ 25.3%

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

      3. associate-*l/ 17.4%

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

      4. *-inverses 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 17.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 19.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 19.3%

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

       3. rem-exp-log 19.3%

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

       4. +-commutative 19.3%

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

   11. Applied egg-rr 19.3%

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

   12. Taylor expanded in th around 0 13.3%

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

   if 1.14999999999999992e-160 < ky < 8.20000000000000008e158 or 2.70000000000000012e205 < 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 40.7%

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

   if 8.20000000000000008e158 < ky < 2.70000000000000012e205

   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/ 99.3%

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

      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 21.6%

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

      1. expm1-log1p-u 21.6%

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

      2. expm1-undefine 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. expm1-define 21.6%

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

      2. associate-*r/ 21.2%

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

      3. associate-*l/ 21.6%

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

      4. *-inverses 21.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 21.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 21.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 21.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 21.8%

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

       3. rem-exp-log 21.8%

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

       4. +-commutative 21.8%

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

   11. Applied egg-rr 21.8%

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

   12. Taylor expanded in th around 0 6.8%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 5.9%

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

       2. expm1-define 5.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 5.7%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 5.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 5.2%

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

       6. sqrt-unprod 5.8%

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

       7. pow2 5.8%

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

   14. Applied egg-rr 5.8%

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

   15. Taylor expanded in th around -inf 45.2%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 45.2%

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

   17. Simplified 45.2%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{+158} \lor \neg \left(ky \leq 2.7 \cdot 10^{+205}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 18.0% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 5.1 \cdot 10^{-184}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{elif}\;ky \leq 6.4 \cdot 10^{+151}:\\ \;\;\;\;\left(th \cdot 2\right) \cdot \frac{1}{th + 2}\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 5.1e-184)
   (+ (+ th 1.0) -1.0)
   (if (<= ky 6.4e+151) (* (* th 2.0) (/ 1.0 (+ th 2.0))) (- th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5.1e-184) {
		tmp = (th + 1.0) + -1.0;
	} else if (ky <= 6.4e+151) {
		tmp = (th * 2.0) * (1.0 / (th + 2.0));
	} else {
		tmp = -th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 5.1d-184) then
        tmp = (th + 1.0d0) + (-1.0d0)
    else if (ky <= 6.4d+151) then
        tmp = (th * 2.0d0) * (1.0d0 / (th + 2.0d0))
    else
        tmp = -th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 5.1e-184) {
		tmp = (th + 1.0) + -1.0;
	} else if (ky <= 6.4e+151) {
		tmp = (th * 2.0) * (1.0 / (th + 2.0));
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 5.1e-184:
		tmp = (th + 1.0) + -1.0
	elif ky <= 6.4e+151:
		tmp = (th * 2.0) * (1.0 / (th + 2.0))
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 5.1e-184)
		tmp = Float64(Float64(th + 1.0) + -1.0);
	elseif (ky <= 6.4e+151)
		tmp = Float64(Float64(th * 2.0) * Float64(1.0 / Float64(th + 2.0)));
	else
		tmp = Float64(-th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 5.1e-184)
		tmp = (th + 1.0) + -1.0;
	elseif (ky <= 6.4e+151)
		tmp = (th * 2.0) * (1.0 / (th + 2.0));
	else
		tmp = -th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 5.1e-184], N[(N[(th + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[ky, 6.4e+151], N[(N[(th * 2.0), $MachinePrecision] * N[(1.0 / N[(th + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-th)]]

Derivation

1. Split input into 3 regimes

2. if ky < 5.0999999999999998e-184

   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/ 89.5%

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

      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 99.6%

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

   5. Taylor expanded in kx around 0 17.4%

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

      1. expm1-log1p-u 17.4%

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

      2. expm1-undefine 18.8%

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

   7. Applied egg-rr 18.8%

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

      1. expm1-define 17.4%

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

      2. associate-*r/ 24.6%

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

      3. associate-*l/ 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th}\right)\right) \]

      4. *-inverses 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 17.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 17.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 18.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 18.8%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 18.8%

          \[\leadsto \color{blue}{\left(1 + \sin th\right)} - 1 \]

       4. +-commutative 18.8%

          \[\leadsto \color{blue}{\left(\sin th + 1\right)} - 1 \]

   11. Applied egg-rr 18.8%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]

   12. Taylor expanded in th around 0 12.6%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]

   if 5.0999999999999998e-184 < ky < 6.39999999999999988e151

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 95.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 95.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 95.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 95.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 95.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 92.9%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 95.7%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 95.7%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   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 37.5%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 37.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 19.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

   7. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 37.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. associate-*r/ 40.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)\right) \]

      3. associate-*l/ 37.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th}\right)\right) \]

      4. *-inverses 37.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 37.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 37.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 19.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 19.5%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 19.5%

          \[\leadsto \color{blue}{\left(1 + \sin th\right)} - 1 \]

       4. +-commutative 19.5%

          \[\leadsto \color{blue}{\left(\sin th + 1\right)} - 1 \]

   11. Applied egg-rr 19.5%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]

   12. Taylor expanded in th around 0 10.3%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. flip-- 10.2%

          \[\leadsto \color{blue}{\frac{\left(1 + th\right) \cdot \left(1 + th\right) - 1 \cdot 1}{\left(1 + th\right) + 1}} \]

       2. div-inv 10.2%

          \[\leadsto \color{blue}{\left(\left(1 + th\right) \cdot \left(1 + th\right) - 1 \cdot 1\right) \cdot \frac{1}{\left(1 + th\right) + 1}} \]

       3. metadata-eval 10.2%

          \[\leadsto \left(\left(1 + th\right) \cdot \left(1 + th\right) - \color{blue}{1}\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       4. difference-of-sqr-1 10.2%

          \[\leadsto \color{blue}{\left(\left(\left(1 + th\right) + 1\right) \cdot \left(\left(1 + th\right) - 1\right)\right)} \cdot \frac{1}{\left(1 + th\right) + 1} \]

       5. +-commutative 10.2%

          \[\leadsto \left(\left(\color{blue}{\left(th + 1\right)} + 1\right) \cdot \left(\left(1 + th\right) - 1\right)\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       6. associate-+l+ 10.2%

          \[\leadsto \left(\color{blue}{\left(th + \left(1 + 1\right)\right)} \cdot \left(\left(1 + th\right) - 1\right)\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       7. metadata-eval 10.2%

          \[\leadsto \left(\left(th + \color{blue}{2}\right) \cdot \left(\left(1 + th\right) - 1\right)\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       8. add-exp-log 9.8%

          \[\leadsto \left(\left(th + 2\right) \cdot \left(\color{blue}{e^{\log \left(1 + th\right)}} - 1\right)\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       9. expm1-define 9.8%

          \[\leadsto \left(\left(th + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)}\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       10. log1p-define 27.8%

           \[\leadsto \left(\left(th + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right)\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       11. expm1-log1p-u 28.2%

           \[\leadsto \left(\left(th + 2\right) \cdot \color{blue}{th}\right) \cdot \frac{1}{\left(1 + th\right) + 1} \]

       12. +-commutative 28.2%

           \[\leadsto \left(\left(th + 2\right) \cdot th\right) \cdot \frac{1}{\color{blue}{\left(th + 1\right)} + 1} \]

       13. associate-+l+ 28.2%

           \[\leadsto \left(\left(th + 2\right) \cdot th\right) \cdot \frac{1}{\color{blue}{th + \left(1 + 1\right)}} \]

       14. metadata-eval 28.2%

           \[\leadsto \left(\left(th + 2\right) \cdot th\right) \cdot \frac{1}{th + \color{blue}{2}} \]

   14. Applied egg-rr 28.2%

       \[\leadsto \color{blue}{\left(\left(th + 2\right) \cdot th\right) \cdot \frac{1}{th + 2}} \]

   15. Taylor expanded in th around 0 29.5%

       \[\leadsto \color{blue}{\left(2 \cdot th\right)} \cdot \frac{1}{th + 2} \]
   16. Step-by-step derivation

       1. *-commutative 29.5%

          \[\leadsto \color{blue}{\left(th \cdot 2\right)} \cdot \frac{1}{th + 2} \]

   17. Simplified 29.5%

       \[\leadsto \color{blue}{\left(th \cdot 2\right)} \cdot \frac{1}{th + 2} \]

   if 6.39999999999999988e151 < 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/ 99.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      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 36.4%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 36.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 23.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

   7. Applied egg-rr 23.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 36.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. associate-*r/ 36.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)\right) \]

      3. associate-*l/ 36.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th}\right)\right) \]

      4. *-inverses 36.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 36.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 36.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 23.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 23.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 23.3%

          \[\leadsto \color{blue}{\left(1 + \sin th\right)} - 1 \]

       4. +-commutative 23.3%

          \[\leadsto \color{blue}{\left(\sin th + 1\right)} - 1 \]

   11. Applied egg-rr 23.3%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]

   12. Taylor expanded in th around 0 5.1%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 4.5%

          \[\leadsto \color{blue}{e^{\log \left(1 + th\right)}} - 1 \]

       2. expm1-define 4.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 17.5%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 12.7%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 12.7%

          \[\leadsto \color{blue}{\sqrt{th} \cdot \sqrt{th}} \]

       6. sqrt-unprod 11.1%

          \[\leadsto \color{blue}{\sqrt{th \cdot th}} \]

       7. pow2 11.1%

          \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   14. Applied egg-rr 11.1%

       \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]

   15. Taylor expanded in th around -inf 28.9%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 28.9%

          \[\leadsto \color{blue}{-th} \]

   17. Simplified 28.9%

       \[\leadsto \color{blue}{-th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 5.1 \cdot 10^{-184}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{elif}\;ky \leq 6.4 \cdot 10^{+151}:\\ \;\;\;\;\left(th \cdot 2\right) \cdot \frac{1}{th + 2}\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 16.9% accurate, 58.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.4e-99) (+ (+ th 1.0) -1.0) (if (<= ky 2.7e-20) th (- th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.4e-99) {
		tmp = (th + 1.0) + -1.0;
	} else if (ky <= 2.7e-20) {
		tmp = th;
	} else {
		tmp = -th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.4d-99) then
        tmp = (th + 1.0d0) + (-1.0d0)
    else if (ky <= 2.7d-20) then
        tmp = th
    else
        tmp = -th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.4e-99) {
		tmp = (th + 1.0) + -1.0;
	} else if (ky <= 2.7e-20) {
		tmp = th;
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.4e-99:
		tmp = (th + 1.0) + -1.0
	elif ky <= 2.7e-20:
		tmp = th
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.4e-99)
		tmp = Float64(Float64(th + 1.0) + -1.0);
	elseif (ky <= 2.7e-20)
		tmp = th;
	else
		tmp = Float64(-th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.4e-99)
		tmp = (th + 1.0) + -1.0;
	elseif (ky <= 2.7e-20)
		tmp = th;
	else
		tmp = -th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.4e-99], N[(N[(th + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[ky, 2.7e-20], th, (-th)]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.4e-99

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 88.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 90.3%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 90.3%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   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 17.6%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 17.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 20.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

   7. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 17.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. associate-*r/ 25.8%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)\right) \]

      3. associate-*l/ 17.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th}\right)\right) \]

      4. *-inverses 17.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 17.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 17.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 20.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 20.0%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 20.0%

          \[\leadsto \color{blue}{\left(1 + \sin th\right)} - 1 \]

       4. +-commutative 20.0%

          \[\leadsto \color{blue}{\left(\sin th + 1\right)} - 1 \]

   11. Applied egg-rr 20.0%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]

   12. Taylor expanded in th around 0 13.7%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]

   if 1.4e-99 < ky < 2.7e-20

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 100.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 100.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 100.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 100.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 91.6%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 99.9%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 83.1%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

   6. Taylor expanded in th around 0 55.6%

      \[\leadsto \color{blue}{th} \]

   if 2.7e-20 < 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/ 99.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      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 33.5%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 33.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 16.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

   7. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 33.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. associate-*r/ 33.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)\right) \]

      3. associate-*l/ 33.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th}\right)\right) \]

      4. *-inverses 33.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 33.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 33.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 16.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 16.8%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 16.8%

          \[\leadsto \color{blue}{\left(1 + \sin th\right)} - 1 \]

       4. +-commutative 16.8%

          \[\leadsto \color{blue}{\left(\sin th + 1\right)} - 1 \]

   11. Applied egg-rr 16.8%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]

   12. Taylor expanded in th around 0 5.9%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 5.3%

          \[\leadsto \color{blue}{e^{\log \left(1 + th\right)}} - 1 \]

       2. expm1-define 5.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 21.8%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 12.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 12.6%

          \[\leadsto \color{blue}{\sqrt{th} \cdot \sqrt{th}} \]

       6. sqrt-unprod 13.7%

          \[\leadsto \color{blue}{\sqrt{th \cdot th}} \]

       7. pow2 13.7%

          \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   14. Applied egg-rr 13.7%

       \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]

   15. Taylor expanded in th around -inf 21.6%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 21.6%

          \[\leadsto \color{blue}{-th} \]

   17. Simplified 21.6%

       \[\leadsto \color{blue}{-th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 13.8% accurate, 101.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 8 \cdot 10^{+110}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;-th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= ky 8e+110) th (- th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8e+110) {
		tmp = th;
	} else {
		tmp = -th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 8d+110) then
        tmp = th
    else
        tmp = -th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 8e+110) {
		tmp = th;
	} else {
		tmp = -th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 8e+110:
		tmp = th
	else:
		tmp = -th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 8e+110)
		tmp = th;
	else
		tmp = Float64(-th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 8e+110)
		tmp = th;
	else
		tmp = -th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 8e+110], th, (-th)]

Derivation

1. Split input into 2 regimes

2. if ky < 8.0000000000000002e110

   1. Initial program 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. unpow2 92.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. sqr-neg 92.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      3. sin-neg 92.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

      4. sin-neg 92.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. unpow2 92.2%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. associate-*l/ 90.2%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      7. associate-/l* 92.2%

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 92.2%

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 23.1%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

   6. Taylor expanded in th around 0 15.9%

      \[\leadsto \color{blue}{th} \]

   if 8.0000000000000002e110 < 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/ 99.4%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

      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 34.5%

      \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 34.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. expm1-undefine 22.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]

   7. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-define 34.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\sin ky}\right)\right)} \]

      2. associate-*r/ 34.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky \cdot \sin th}{\sin ky}}\right)\right) \]

      3. associate-*l/ 34.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin ky}{\sin ky} \cdot \sin th}\right)\right) \]

      4. *-inverses 34.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} \cdot \sin th\right)\right) \]

      5. *-lft-identity 34.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin th}\right)\right) \]

   9. Simplified 34.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin th\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 22.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin th\right)} - 1} \]

       2. log1p-undefine 22.4%

          \[\leadsto e^{\color{blue}{\log \left(1 + \sin th\right)}} - 1 \]

       3. rem-exp-log 22.4%

          \[\leadsto \color{blue}{\left(1 + \sin th\right)} - 1 \]

       4. +-commutative 22.4%

          \[\leadsto \color{blue}{\left(\sin th + 1\right)} - 1 \]

   11. Applied egg-rr 22.4%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]

   12. Taylor expanded in th around 0 6.7%

       \[\leadsto \color{blue}{\left(1 + th\right)} - 1 \]
   13. Step-by-step derivation

       1. add-exp-log 5.9%

          \[\leadsto \color{blue}{e^{\log \left(1 + th\right)}} - 1 \]

       2. expm1-define 5.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(1 + th\right)\right)} \]

       3. log1p-define 17.7%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(th\right)}\right) \]

       4. add-sqr-sqrt 10.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{th} \cdot \sqrt{th}}\right)\right) \]

       5. expm1-log1p-u 10.8%

          \[\leadsto \color{blue}{\sqrt{th} \cdot \sqrt{th}} \]

       6. sqrt-unprod 9.7%

          \[\leadsto \color{blue}{\sqrt{th \cdot th}} \]

       7. pow2 9.7%

          \[\leadsto \sqrt{\color{blue}{{th}^{2}}} \]

   14. Applied egg-rr 9.7%

       \[\leadsto \color{blue}{\sqrt{{th}^{2}}} \]

   15. Taylor expanded in th around -inf 24.9%

       \[\leadsto \color{blue}{-1 \cdot th} \]
   16. Step-by-step derivation

       1. neg-mul-1 24.9%

          \[\leadsto \color{blue}{-th} \]

   17. Simplified 24.9%

       \[\leadsto \color{blue}{-th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 13.8% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.4%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. unpow2 93.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

   2. sqr-neg 93.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(-\sin kx\right) \cdot \left(-\sin kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   3. sin-neg 93.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin \left(-kx\right)} \cdot \left(-\sin kx\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. sin-neg 93.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\sin \left(-kx\right) \cdot \color{blue}{\sin \left(-kx\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. unpow2 93.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin \left(-kx\right)}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

   6. associate-*l/ 91.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   7. associate-/l* 93.3%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin \left(-kx\right)}^{2} + {\sin ky}^{2}}}} \]

   8. unpow2 93.3%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin \left(-kx\right) \cdot \sin \left(-kx\right)} + {\sin ky}^{2}}} \]

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 24.9%

   \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin ky}} \]

6. Taylor expanded in th around 0 16.3%

   \[\leadsto \color{blue}{th} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))