Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 19.9s

Alternatives: 14

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% 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 93.9%

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

   1. remove-double-neg 93.9%

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

   2. sin-neg 93.9%

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

   3. neg-mul-1 93.9%

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

   4. *-commutative 93.9%

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

   5. associate-*l* 93.9%

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

   6. associate-*l/ 91.4%

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

   7. associate-/r/ 91.4%

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

   8. associate-*l/ 93.9%

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

   9. associate-/r/ 93.9%

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

   10. sin-neg 93.9%

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

   11. neg-mul-1 93.9%

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

   12. associate-/r* 93.9%

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

   13. associate-/r/ 93.9%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-132} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-121}\right) \land \sin ky \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (/ 1.0 (sqrt (pow (sin th) -2.0)))
   (if (or (<= (sin ky) 5e-132)
           (and (not (<= (sin ky) 5e-121)) (<= (sin ky) 5e-44)))
     (* (sin th) (fabs (/ (sin ky) (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if ((sin(ky) <= 5e-132) || (!(sin(ky) <= 5e-121) && (sin(ky) <= 5e-44))) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.002d0)) then
        tmp = 1.0d0 / sqrt((sin(th) ** (-2.0d0)))
    else if ((sin(ky) <= 5d-132) .or. (.not. (sin(ky) <= 5d-121)) .and. (sin(ky) <= 5d-44)) then
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.002) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if ((Math.sin(ky) <= 5e-132) || (!(Math.sin(ky) <= 5e-121) && (Math.sin(ky) <= 5e-44))) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.002:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif (math.sin(ky) <= 5e-132) or (not (math.sin(ky) <= 5e-121) and (math.sin(ky) <= 5e-44)):
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif ((sin(ky) <= 5e-132) || (!(sin(ky) <= 5e-121) && (sin(ky) <= 5e-44)))
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.002)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif ((sin(ky) <= 5e-132) || (~((sin(ky) <= 5e-121)) && (sin(ky) <= 5e-44)))
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 5e-132], And[N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 5e-121]], $MachinePrecision], LessEqual[N[Sin[ky], $MachinePrecision], 5e-44]]], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. sin-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/r/ 99.6%

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

      8. associate-*l/ 99.8%

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

      9. associate-/r/ 99.8%

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

      10. sin-neg 99.8%

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

      11. neg-mul-1 99.8%

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

      12. associate-/r* 99.8%

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

      13. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.4%

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

      3. hypot-udef 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. hypot-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in kx around 0 2.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 19.8%

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

      3. inv-pow 19.8%

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

      4. inv-pow 19.8%

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

      5. pow-prod-up 19.8%

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

      6. metadata-eval 19.8%

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

   9. Applied egg-rr 19.8%

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

   if -2e-3 < (sin.f64 ky) < 4.9999999999999999e-132 or 4.99999999999999989e-121 < (sin.f64 ky) < 5.00000000000000039e-44

   1. Initial program 87.0%

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

   3. Taylor expanded in ky around 0 41.8%

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

      1. add-sqr-sqrt 19.5%

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

      2. sqrt-unprod 33.7%

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

      3. pow2 33.7%

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

   5. Applied egg-rr 33.7%

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

      1. unpow2 33.7%

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

      2. rem-sqrt-square 41.7%

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

   7. Simplified 41.7%

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

   if 4.9999999999999999e-132 < (sin.f64 ky) < 4.99999999999999989e-121 or 5.00000000000000039e-44 < (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. remove-double-neg 99.7%

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

      2. sin-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. associate-*l/ 98.6%

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

      7. associate-/r/ 98.6%

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

      8. associate-*l/ 99.7%

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

      9. associate-/r/ 99.6%

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

      10. sin-neg 99.6%

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

      11. neg-mul-1 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 56.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-132} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-121}\right) \land \sin ky \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-135} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-121}\right) \land \sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (/ 1.0 (sqrt (pow (sin th) -2.0)))
   (if (or (<= (sin ky) 1e-135)
           (and (not (<= (sin ky) 5e-121)) (<= (sin ky) 5e-99)))
     (fabs (* (sin th) (/ ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if ((sin(ky) <= 1e-135) || (!(sin(ky) <= 5e-121) && (sin(ky) <= 5e-99))) {
		tmp = fabs((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) <= (-0.002d0)) then
        tmp = 1.0d0 / sqrt((sin(th) ** (-2.0d0)))
    else if ((sin(ky) <= 1d-135) .or. (.not. (sin(ky) <= 5d-121)) .and. (sin(ky) <= 5d-99)) then
        tmp = abs((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) <= -0.002) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if ((Math.sin(ky) <= 1e-135) || (!(Math.sin(ky) <= 5e-121) && (Math.sin(ky) <= 5e-99))) {
		tmp = Math.abs((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) <= -0.002:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif (math.sin(ky) <= 1e-135) or (not (math.sin(ky) <= 5e-121) and (math.sin(ky) <= 5e-99)):
		tmp = math.fabs((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) <= -0.002)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif ((sin(ky) <= 1e-135) || (!(sin(ky) <= 5e-121) && (sin(ky) <= 5e-99)))
		tmp = abs(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) <= -0.002)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif ((sin(ky) <= 1e-135) || (~((sin(ky) <= 5e-121)) && (sin(ky) <= 5e-99)))
		tmp = abs((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], -0.002], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[Sin[ky], $MachinePrecision], 1e-135], And[N[Not[LessEqual[N[Sin[ky], $MachinePrecision], 5e-121]], $MachinePrecision], LessEqual[N[Sin[ky], $MachinePrecision], 5e-99]]], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. sin-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/r/ 99.6%

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

      8. associate-*l/ 99.8%

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

      9. associate-/r/ 99.8%

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

      10. sin-neg 99.8%

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

      11. neg-mul-1 99.8%

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

      12. associate-/r* 99.8%

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

      13. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.4%

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

      3. hypot-udef 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. hypot-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in kx around 0 2.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 19.8%

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

      3. inv-pow 19.8%

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

      4. inv-pow 19.8%

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

      5. pow-prod-up 19.8%

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

      6. metadata-eval 19.8%

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

   9. Applied egg-rr 19.8%

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

   if -2e-3 < (sin.f64 ky) < 1e-135 or 4.99999999999999989e-121 < (sin.f64 ky) < 4.99999999999999969e-99

   1. Initial program 86.2%

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

   3. Taylor expanded in ky around 0 40.8%

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

   4. Taylor expanded in ky around inf 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/l* 40.8%

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

      3. associate-/r/ 40.8%

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

   6. Simplified 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*r/ 37.7%

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

      3. associate-*l/ 40.8%

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

      4. rem-log-exp 22.7%

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

      5. add-sqr-sqrt 22.5%

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

      6. sqrt-unprod 23.0%

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

      7. pow2 23.0%

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

      8. *-commutative 23.0%

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

      9. rem-log-exp 31.8%

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

   8. Applied egg-rr 31.8%

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

      1. unpow2 31.8%

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

      2. rem-sqrt-square 50.2%

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

   10. Simplified 50.2%

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

   11. Taylor expanded in ky around 0 50.2%

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

   if 1e-135 < (sin.f64 ky) < 4.99999999999999989e-121 or 4.99999999999999969e-99 < (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. remove-double-neg 99.7%

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

      2. sin-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. associate-*l/ 98.7%

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

      7. associate-/r/ 98.7%

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

      8. associate-*l/ 99.7%

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

      9. associate-/r/ 99.6%

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

      10. sin-neg 99.6%

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

      11. neg-mul-1 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 54.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-135} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-121}\right) \land \sin ky \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.48:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq -0.01:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.48)
   (* (sin th) (fabs (/ (sin ky) (sin kx))))
   (if (<= (sin th) -0.01)
     (sin th)
     (if (<= (sin th) 0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (* (sin ky) (fabs (/ (sin th) (sin kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.48) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else if (sin(th) <= -0.01) {
		tmp = sin(th);
	} else if (sin(th) <= 0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.48) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else if (Math.sin(th) <= -0.01) {
		tmp = Math.sin(th);
	} else if (Math.sin(th) <= 0.02) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.48:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	elif math.sin(th) <= -0.01:
		tmp = math.sin(th)
	elif math.sin(th) <= 0.02:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.48)
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	elseif (sin(th) <= -0.01)
		tmp = sin(th);
	elseif (sin(th) <= 0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.48)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	elseif (sin(th) <= -0.01)
		tmp = sin(th);
	elseif (sin(th) <= 0.02)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.48], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], -0.01], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 th) < -0.47999999999999998

   1. Initial program 97.5%

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

   3. Taylor expanded in ky around 0 14.4%

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

      1. add-sqr-sqrt 4.8%

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

      2. sqrt-unprod 13.8%

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

      3. pow2 13.8%

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

   5. Applied egg-rr 13.8%

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

      1. unpow2 13.8%

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

      2. rem-sqrt-square 21.6%

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

   7. Simplified 21.6%

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

   if -0.47999999999999998 < (sin.f64 th) < -0.0100000000000000002

   1. Initial program 92.2%

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

      1. remove-double-neg 92.2%

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

      2. sin-neg 92.2%

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

      3. neg-mul-1 92.2%

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

      4. *-commutative 92.2%

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

      5. associate-*l* 92.2%

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

      6. associate-*l/ 92.3%

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

      7. associate-/r/ 92.3%

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

      8. associate-*l/ 92.2%

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

      9. associate-/r/ 92.3%

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

      10. sin-neg 92.3%

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

      11. neg-mul-1 92.3%

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

      12. associate-/r* 92.3%

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

      13. associate-/r/ 92.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 28.5%

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

   if -0.0100000000000000002 < (sin.f64 th) < 0.0200000000000000004

   1. Initial program 92.4%

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

      1. remove-double-neg 92.4%

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

      2. sin-neg 92.4%

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

      3. neg-mul-1 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*l* 92.4%

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

      6. associate-*l/ 87.6%

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

      7. associate-/r/ 87.6%

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

      8. associate-*l/ 92.4%

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

      9. associate-/r/ 92.4%

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

      10. sin-neg 92.4%

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

      11. neg-mul-1 92.4%

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

      12. associate-/r* 92.4%

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

      13. associate-/r/ 92.4%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-udef 22.7%

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

      3. hypot-udef 21.2%

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

      4. unpow2 21.2%

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

      5. unpow2 21.2%

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

      6. +-commutative 21.2%

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

      7. unpow2 21.2%

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

      8. unpow2 21.2%

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

      9. hypot-def 22.7%

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

   6. Applied egg-rr 22.7%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-/r/ 99.7%

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

      4. hypot-def 92.4%

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

      5. unpow2 92.4%

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

      6. unpow2 92.4%

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

      7. +-commutative 92.4%

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

      8. unpow2 92.4%

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

      9. unpow2 92.4%

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

      10. hypot-def 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in th around 0 91.0%

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

       1. associate-*l/ 91.2%

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

       2. +-commutative 91.2%

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

       3. unpow2 91.2%

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

       4. unpow2 91.2%

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

       5. hypot-def 98.4%

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

       6. *-lft-identity 98.4%

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

       7. hypot-def 91.2%

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

       8. unpow2 91.2%

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

       9. unpow2 91.2%

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

       10. +-commutative 91.2%

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

       11. unpow2 91.2%

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

       12. unpow2 91.2%

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

       13. hypot-def 98.4%

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

   11. Simplified 98.4%

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

   if 0.0200000000000000004 < (sin.f64 th)

   1. Initial program 95.0%

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

   3. Taylor expanded in ky around 0 20.5%

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

   4. Taylor expanded in ky around inf 20.4%

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

      1. *-commutative 20.4%

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

      2. associate-/l* 20.5%

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

      3. associate-/r/ 20.4%

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

   6. Simplified 20.4%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 40.1%

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

      3. pow2 40.1%

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

   8. Applied egg-rr 40.1%

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

      1. unpow2 40.1%

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

      2. rem-sqrt-square 42.2%

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

   10. Simplified 42.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.48:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin th \leq -0.01:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{if}\;\sin kx \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\left|t_1\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (sin kx)))))
   (if (<= (sin kx) 5e-306)
     (fabs t_1)
     (if (<= (sin kx) 5e-19) (/ 1.0 (/ (sin ky) (* (sin ky) (sin th)))) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / sin(kx));
	double tmp;
	if (sin(kx) <= 5e-306) {
		tmp = fabs(t_1);
	} else if (sin(kx) <= 5e-19) {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	} else {
		tmp = 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(ky) / sin(kx))
    if (sin(kx) <= 5d-306) then
        tmp = abs(t_1)
    else if (sin(kx) <= 5d-19) then
        tmp = 1.0d0 / (sin(ky) / (sin(ky) * sin(th)))
    else
        tmp = 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(ky) / Math.sin(kx));
	double tmp;
	if (Math.sin(kx) <= 5e-306) {
		tmp = Math.abs(t_1);
	} else if (Math.sin(kx) <= 5e-19) {
		tmp = 1.0 / (Math.sin(ky) / (Math.sin(ky) * Math.sin(th)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) * (math.sin(ky) / math.sin(kx))
	tmp = 0
	if math.sin(kx) <= 5e-306:
		tmp = math.fabs(t_1)
	elif math.sin(kx) <= 5e-19:
		tmp = 1.0 / (math.sin(ky) / (math.sin(ky) * math.sin(th)))
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / sin(kx)))
	tmp = 0.0
	if (sin(kx) <= 5e-306)
		tmp = abs(t_1);
	elseif (sin(kx) <= 5e-19)
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) * (sin(ky) / sin(kx));
	tmp = 0.0;
	if (sin(kx) <= 5e-306)
		tmp = abs(t_1);
	elseif (sin(kx) <= 5e-19)
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-306], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-19], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < 4.99999999999999998e-306

   1. Initial program 94.7%

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

   3. Taylor expanded in ky around 0 12.9%

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

   4. Taylor expanded in ky around inf 13.0%

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

      1. *-commutative 13.0%

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

      2. associate-/l* 12.9%

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

      3. associate-/r/ 12.9%

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

   6. Simplified 12.9%

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

      1. *-commutative 12.9%

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

      2. associate-*r/ 13.0%

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

      3. associate-*l/ 12.9%

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

      4. rem-log-exp 13.1%

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

      5. add-sqr-sqrt 12.6%

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

      6. sqrt-unprod 15.4%

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

      7. pow2 15.4%

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

      8. *-commutative 15.4%

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

      9. rem-log-exp 21.1%

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

   8. Applied egg-rr 21.1%

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

      1. unpow2 21.1%

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

      2. rem-sqrt-square 29.0%

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

   10. Simplified 29.0%

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

   if 4.99999999999999998e-306 < (sin.f64 kx) < 5.0000000000000004e-19

   1. Initial program 86.8%

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

      1. remove-double-neg 86.8%

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

      2. sin-neg 86.8%

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

      3. neg-mul-1 86.8%

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

      4. *-commutative 86.8%

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

      5. associate-*l* 86.8%

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

      6. associate-*l/ 82.4%

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

      7. associate-/r/ 82.4%

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

      8. associate-*l/ 86.8%

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

      9. associate-/r/ 86.7%

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

      10. sin-neg 86.7%

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

      11. neg-mul-1 86.7%

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

      12. associate-/r* 86.7%

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

      13. associate-/r/ 86.8%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 92.5%

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

      2. clear-num 92.1%

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

      3. hypot-udef 82.0%

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

      4. unpow2 82.0%

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

      5. unpow2 82.0%

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

      6. +-commutative 82.0%

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

      7. unpow2 82.0%

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

      8. unpow2 82.0%

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

      9. hypot-def 92.1%

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

   6. Applied egg-rr 92.1%

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

   7. Taylor expanded in kx around 0 38.0%

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

   if 5.0000000000000004e-19 < (sin.f64 kx)

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0 55.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 5e-306)
   (fabs (* (sin ky) (/ (sin th) (sin kx))))
   (if (<= (sin kx) 5e-19)
     (/ 1.0 (/ (sin ky) (* (sin ky) (sin th))))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 5e-306) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 5e-19) {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	} else {
		tmp = sin(th) * (sin(ky) / 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(kx) <= 5d-306) then
        tmp = abs((sin(ky) * (sin(th) / sin(kx))))
    else if (sin(kx) <= 5d-19) then
        tmp = 1.0d0 / (sin(ky) / (sin(ky) * sin(th)))
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= 5e-306) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 5e-19) {
		tmp = 1.0 / (Math.sin(ky) / (Math.sin(ky) * Math.sin(th)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= 5e-306:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 5e-19:
		tmp = 1.0 / (math.sin(ky) / (math.sin(ky) * math.sin(th)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 5e-306)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 5e-19)
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= 5e-306)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 5e-19)
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-306], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-19], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < 4.99999999999999998e-306

   1. Initial program 94.7%

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

   3. Taylor expanded in ky around 0 12.9%

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

      1. add-sqr-sqrt 11.8%

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

      2. sqrt-unprod 21.1%

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

      3. pow2 21.1%

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

      4. *-commutative 21.1%

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

   5. Applied egg-rr 21.1%

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

      1. unpow2 21.1%

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

      2. rem-sqrt-square 29.0%

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

      3. associate-*r/ 27.8%

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

      4. associate-/l* 29.0%

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

      5. associate-/r/ 29.1%

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

   7. Simplified 29.1%

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

   if 4.99999999999999998e-306 < (sin.f64 kx) < 5.0000000000000004e-19

   1. Initial program 86.8%

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

      1. remove-double-neg 86.8%

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

      2. sin-neg 86.8%

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

      3. neg-mul-1 86.8%

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

      4. *-commutative 86.8%

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

      5. associate-*l* 86.8%

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

      6. associate-*l/ 82.4%

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

      7. associate-/r/ 82.4%

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

      8. associate-*l/ 86.8%

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

      9. associate-/r/ 86.7%

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

      10. sin-neg 86.7%

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

      11. neg-mul-1 86.7%

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

      12. associate-/r* 86.7%

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

      13. associate-/r/ 86.8%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 92.5%

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

      2. clear-num 92.1%

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

      3. hypot-udef 82.0%

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

      4. unpow2 82.0%

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

      5. unpow2 82.0%

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

      6. +-commutative 82.0%

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

      7. unpow2 82.0%

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

      8. unpow2 82.0%

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

      9. hypot-def 92.1%

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

   6. Applied egg-rr 92.1%

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

   7. Taylor expanded in kx around 0 38.0%

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

   if 5.0000000000000004e-19 < (sin.f64 kx)

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0 55.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (fabs (sin th))
   (if (<= (sin ky) 5e-139) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-139) {
		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) <= (-0.002d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-139) 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) <= -0.002) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-139) {
		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) <= -0.002:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-139:
		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) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-139)
		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) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-139)
		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], -0.002], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-139], 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) < -2e-3

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. sin-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/r/ 99.6%

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

      8. associate-*l/ 99.8%

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

      9. associate-/r/ 99.8%

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

      10. sin-neg 99.8%

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

      11. neg-mul-1 99.8%

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

      12. associate-/r* 99.8%

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

      13. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.4%

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

      3. hypot-udef 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. hypot-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in kx around 0 2.4%

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

      1. remove-double-div 2.4%

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

      2. add-sqr-sqrt 1.5%

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

      3. sqrt-unprod 19.8%

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

      4. pow2 19.8%

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

   9. Applied egg-rr 19.8%

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

       1. unpow2 19.8%

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

       2. rem-sqrt-square 26.2%

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

   11. Simplified 26.2%

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

   if -2e-3 < (sin.f64 ky) < 5.00000000000000034e-139

   1. Initial program 85.7%

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

   3. Taylor expanded in ky around 0 41.3%

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

   if 5.00000000000000034e-139 < (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. remove-double-neg 99.7%

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

      2. sin-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. associate-*l/ 98.7%

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

      7. associate-/r/ 98.7%

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

      8. associate-*l/ 99.7%

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

      9. associate-/r/ 99.6%

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

      10. sin-neg 99.6%

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

      11. neg-mul-1 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 53.7%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (/ 1.0 (sqrt (pow (sin th) -2.0)))
   (if (<= (sin ky) 5e-139) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= 5e-139) {
		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) <= (-0.002d0)) then
        tmp = 1.0d0 / sqrt((sin(th) ** (-2.0d0)))
    else if (sin(ky) <= 5d-139) 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) <= -0.002) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= 5e-139) {
		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) <= -0.002:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= 5e-139:
		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) <= -0.002)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= 5e-139)
		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) <= -0.002)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= 5e-139)
		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], -0.002], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-139], 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) < -2e-3

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. sin-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/r/ 99.6%

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

      8. associate-*l/ 99.8%

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

      9. associate-/r/ 99.8%

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

      10. sin-neg 99.8%

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

      11. neg-mul-1 99.8%

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

      12. associate-/r* 99.8%

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

      13. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.4%

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

      3. hypot-udef 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. hypot-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in kx around 0 2.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 19.8%

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

      3. inv-pow 19.8%

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

      4. inv-pow 19.8%

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

      5. pow-prod-up 19.8%

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

      6. metadata-eval 19.8%

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

   9. Applied egg-rr 19.8%

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

   if -2e-3 < (sin.f64 ky) < 5.00000000000000034e-139

   1. Initial program 85.7%

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

   3. Taylor expanded in ky around 0 41.3%

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

   if 5.00000000000000034e-139 < (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. remove-double-neg 99.7%

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

      2. sin-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. associate-*l/ 98.7%

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

      7. associate-/r/ 98.7%

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

      8. associate-*l/ 99.7%

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

      9. associate-/r/ 99.6%

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

      10. sin-neg 99.6%

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

      11. neg-mul-1 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 53.7%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;th \leq 0.21:\\ \;\;\;\;\frac{\sin ky}{\frac{t_1}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky))))
   (if (<= th 0.21) (/ (sin ky) (/ t_1 th)) (/ 1.0 (/ t_1 (* ky (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double tmp;
	if (th <= 0.21) {
		tmp = sin(ky) / (t_1 / th);
	} else {
		tmp = 1.0 / (t_1 / (ky * sin(th)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (th <= 0.21) {
		tmp = Math.sin(ky) / (t_1 / th);
	} else {
		tmp = 1.0 / (t_1 / (ky * Math.sin(th)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if th <= 0.21:
		tmp = math.sin(ky) / (t_1 / th)
	else:
		tmp = 1.0 / (t_1 / (ky * math.sin(th)))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (th <= 0.21)
		tmp = Float64(sin(ky) / Float64(t_1 / th));
	else
		tmp = Float64(1.0 / Float64(t_1 / Float64(ky * sin(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (th <= 0.21)
		tmp = sin(ky) / (t_1 / th);
	else
		tmp = 1.0 / (t_1 / (ky * sin(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[th, 0.21], N[(N[Sin[ky], $MachinePrecision] / N[(t$95$1 / th), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 0.209999999999999992

   1. Initial program 93.4%

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

      1. remove-double-neg 93.4%

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

      2. sin-neg 93.4%

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

      3. neg-mul-1 93.4%

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

      4. *-commutative 93.4%

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

      5. associate-*l* 93.4%

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

      6. associate-*l/ 90.2%

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

      7. associate-/r/ 90.2%

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

      8. associate-*l/ 93.4%

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

      9. associate-/r/ 93.3%

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

      10. sin-neg 93.3%

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

      11. neg-mul-1 93.3%

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

      12. associate-/r* 93.3%

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

      13. associate-/r/ 93.4%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-udef 36.3%

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

      3. hypot-udef 35.2%

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

      4. unpow2 35.2%

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

      5. unpow2 35.2%

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

      6. +-commutative 35.2%

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

      7. unpow2 35.2%

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

      8. unpow2 35.2%

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

      9. hypot-def 36.3%

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

   6. Applied egg-rr 36.3%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-/r/ 99.6%

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

      4. hypot-def 93.3%

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

      5. unpow2 93.3%

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

      6. unpow2 93.3%

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

      7. +-commutative 93.3%

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

      8. unpow2 93.3%

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

      9. unpow2 93.3%

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

      10. hypot-def 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in th around 0 61.4%

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

       1. associate-*l/ 61.5%

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

       2. +-commutative 61.5%

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

       3. unpow2 61.5%

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

       4. unpow2 61.5%

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

       5. hypot-def 66.3%

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

       6. *-lft-identity 66.3%

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

       7. hypot-def 61.5%

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

       8. unpow2 61.5%

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

       9. unpow2 61.5%

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

       10. +-commutative 61.5%

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

       11. unpow2 61.5%

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

       12. unpow2 61.5%

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

       13. hypot-def 66.3%

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

   11. Simplified 66.3%

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

   if 0.209999999999999992 < th

   1. Initial program 95.9%

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

      1. remove-double-neg 95.9%

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

      2. sin-neg 95.9%

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

      3. neg-mul-1 95.9%

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

      4. *-commutative 95.9%

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

      5. associate-*l* 95.9%

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

      6. associate-*l/ 95.9%

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

      7. associate-/r/ 95.9%

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

      8. associate-*l/ 95.9%

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

      9. associate-/r/ 95.9%

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

      10. sin-neg 95.9%

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

      11. neg-mul-1 95.9%

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

      12. associate-/r* 95.9%

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

      13. associate-/r/ 95.9%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

      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. hypot-udef 95.7%

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

      4. unpow2 95.7%

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

      5. unpow2 95.7%

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

      6. +-commutative 95.7%

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

      7. unpow2 95.7%

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

      8. unpow2 95.7%

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

      9. hypot-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ky around 0 54.2%

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{ky \cdot \sin th}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.21:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (fabs (sin th))
   (if (<= (sin ky) 5e-139) (/ ky (/ kx (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-139) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.002d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-139) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.002) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-139) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.002:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-139:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-139)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.002)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-139)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-139], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. sin-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/r/ 99.6%

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

      8. associate-*l/ 99.8%

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

      9. associate-/r/ 99.8%

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

      10. sin-neg 99.8%

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

      11. neg-mul-1 99.8%

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

      12. associate-/r* 99.8%

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

      13. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.6%

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

      2. clear-num 99.4%

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

      3. hypot-udef 99.4%

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

      4. unpow2 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. hypot-def 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in kx around 0 2.4%

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

      1. remove-double-div 2.4%

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

      2. add-sqr-sqrt 1.5%

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

      3. sqrt-unprod 19.8%

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

      4. pow2 19.8%

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

   9. Applied egg-rr 19.8%

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

       1. unpow2 19.8%

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

       2. rem-sqrt-square 26.2%

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

   11. Simplified 26.2%

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

   if -2e-3 < (sin.f64 ky) < 5.00000000000000034e-139

   1. Initial program 85.7%

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

   3. Taylor expanded in ky around 0 41.3%

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

   4. Taylor expanded in kx around 0 27.4%

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

      1. associate-/l* 30.5%

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

   6. Simplified 30.5%

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

   7. Taylor expanded in ky around 0 27.4%

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

      1. associate-/l* 30.5%

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

   9. Simplified 30.5%

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

   if 5.00000000000000034e-139 < (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. remove-double-neg 99.7%

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

      2. sin-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. associate-*l/ 98.7%

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

      7. associate-/r/ 98.7%

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

      8. associate-*l/ 99.7%

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

      9. associate-/r/ 99.6%

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

      10. sin-neg 99.6%

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

      11. neg-mul-1 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-/r/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 53.7%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.02 \cdot 10^{-138}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.02e-138) (/ ky (/ kx (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.02e-138) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 1.02d-138) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.02e-138) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.02e-138:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.02e-138)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.02e-138)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.02e-138], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.02000000000000007e-138

   1. Initial program 90.9%

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

   3. Taylor expanded in ky around 0 28.7%

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

   4. Taylor expanded in kx around 0 19.0%

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

      1. associate-/l* 21.0%

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

   6. Simplified 21.0%

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

   7. Taylor expanded in ky around 0 18.6%

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

      1. associate-/l* 20.6%

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

   9. Simplified 20.6%

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

   if 1.02000000000000007e-138 < 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. remove-double-neg 99.6%

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

      2. sin-neg 99.6%

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

      3. neg-mul-1 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l* 99.6%

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

      6. associate-*l/ 98.6%

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

      7. associate-/r/ 98.6%

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

      8. associate-*l/ 99.6%

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

      9. associate-/r/ 99.7%

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

      10. sin-neg 99.7%

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

      11. neg-mul-1 99.7%

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

      12. associate-/r* 99.7%

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

      13. associate-/r/ 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 35.6%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.02 \cdot 10^{-138}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.8% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. remove-double-neg 93.9%

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

   2. sin-neg 93.9%

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

   3. neg-mul-1 93.9%

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

   4. *-commutative 93.9%

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

   5. associate-*l* 93.9%

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

   6. associate-*l/ 91.4%

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

   7. associate-/r/ 91.4%

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

   8. associate-*l/ 93.9%

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

   9. associate-/r/ 93.9%

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

   10. sin-neg 93.9%

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

   11. neg-mul-1 93.9%

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

   12. associate-/r* 93.9%

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

   13. associate-/r/ 93.9%

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

3. Simplified 99.7%

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

5. Taylor expanded in kx around 0 23.5%

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

6. Final simplification 23.5%

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

Alternative 13: 14.9% accurate, 78.8× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / 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 = 1.0d0 / ((th * 0.16666666666666666d0) + (1.0d0 / th))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. remove-double-neg 93.9%

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

   2. sin-neg 93.9%

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

   3. neg-mul-1 93.9%

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

   4. *-commutative 93.9%

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

   5. associate-*l* 93.9%

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

   6. associate-*l/ 91.4%

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

   7. associate-/r/ 91.4%

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

   8. associate-*l/ 93.9%

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

   9. associate-/r/ 93.9%

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

   10. sin-neg 93.9%

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

   11. neg-mul-1 93.9%

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

   12. associate-/r* 93.9%

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

   13. associate-/r/ 93.9%

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

3. Simplified 99.7%

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

   1. associate-*l/ 96.0%

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

   2. clear-num 95.4%

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

   3. hypot-udef 90.8%

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

   4. unpow2 90.8%

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

   5. unpow2 90.8%

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

   6. +-commutative 90.8%

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

   7. unpow2 90.8%

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

   8. unpow2 90.8%

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

   9. hypot-def 95.4%

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

6. Applied egg-rr 95.4%

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

7. Taylor expanded in kx around 0 23.5%

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

8. Taylor expanded in th around 0 14.2%

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

9. Final simplification 14.2%

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

Alternative 14: 14.2% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.9%

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

   1. remove-double-neg 93.9%

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

   2. sin-neg 93.9%

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

   3. neg-mul-1 93.9%

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

   4. *-commutative 93.9%

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

   5. associate-*l* 93.9%

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

   6. associate-*l/ 91.4%

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

   7. associate-/r/ 91.4%

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

   8. associate-*l/ 93.9%

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

   9. associate-/r/ 93.9%

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

   10. sin-neg 93.9%

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

   11. neg-mul-1 93.9%

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

   12. associate-/r* 93.9%

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

   13. associate-/r/ 93.9%

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

3. Simplified 99.7%

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

   1. associate-*l/ 96.0%

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

   2. clear-num 95.4%

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

   3. hypot-udef 90.8%

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

   4. unpow2 90.8%

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

   5. unpow2 90.8%

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

   6. +-commutative 90.8%

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

   7. unpow2 90.8%

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

   8. unpow2 90.8%

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

   9. hypot-def 95.4%

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

6. Applied egg-rr 95.4%

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

7. Taylor expanded in kx around 0 23.5%

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

8. Taylor expanded in th around 0 13.4%

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

9. Final simplification 13.4%

   \[\leadsto th \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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)))