Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 15.5s

Alternatives: 21

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: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. unpow2 92.3%

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

   2. sqr-neg 92.3%

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

   3. sin-neg 92.3%

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

   4. sin-neg 92.3%

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

   5. unpow2 92.3%

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

   6. associate-*l/ 89.6%

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

   7. associate-/l* 92.2%

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

   8. unpow2 92.2%

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

3. Simplified 99.6%

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

   1. associate-*r/ 93.9%

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

   2. hypot-undefine 89.6%

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

   3. unpow2 89.6%

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

   4. unpow2 89.6%

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

   5. +-commutative 89.6%

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

   6. associate-*l/ 92.3%

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

   7. *-commutative 92.3%

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

   8. clear-num 92.3%

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

   9. un-div-inv 92.3%

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

   10. +-commutative 92.3%

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

   11. unpow2 92.3%

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

   12. unpow2 92.3%

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

   13. hypot-undefine 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 2: 48.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot ky\\ \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2.5 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-98} \lor \neg \left(\sin kx \leq 2 \cdot 10^{-60}\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) ky)))
   (if (<= (sin kx) -0.005)
     (* t_1 (sqrt (/ 1.0 (- 0.5 (* 0.5 (cos (* kx 2.0)))))))
     (if (<= (sin kx) 2.5e-119)
       (sin th)
       (if (or (<= (sin kx) 1e-98) (not (<= (sin kx) 2e-60)))
         (* (sin th) (/ (sin ky) (sin kx)))
         (/ t_1 ky))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * ky;
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = t_1 * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	} else if (sin(kx) <= 2.5e-119) {
		tmp = sin(th);
	} else if ((sin(kx) <= 1e-98) || !(sin(kx) <= 2e-60)) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = t_1 / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) * ky
    if (sin(kx) <= (-0.005d0)) then
        tmp = t_1 * sqrt((1.0d0 / (0.5d0 - (0.5d0 * cos((kx * 2.0d0))))))
    else if (sin(kx) <= 2.5d-119) then
        tmp = sin(th)
    else if ((sin(kx) <= 1d-98) .or. (.not. (sin(kx) <= 2d-60))) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else
        tmp = t_1 / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) * ky;
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = t_1 * Math.sqrt((1.0 / (0.5 - (0.5 * Math.cos((kx * 2.0))))));
	} else if (Math.sin(kx) <= 2.5e-119) {
		tmp = Math.sin(th);
	} else if ((Math.sin(kx) <= 1e-98) || !(Math.sin(kx) <= 2e-60)) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = t_1 / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) * ky
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = t_1 * math.sqrt((1.0 / (0.5 - (0.5 * math.cos((kx * 2.0))))))
	elif math.sin(kx) <= 2.5e-119:
		tmp = math.sin(th)
	elif (math.sin(kx) <= 1e-98) or not (math.sin(kx) <= 2e-60):
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = t_1 / ky
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * ky)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(t_1 * sqrt(Float64(1.0 / Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0)))))));
	elseif (sin(kx) <= 2.5e-119)
		tmp = sin(th);
	elseif ((sin(kx) <= 1e-98) || !(sin(kx) <= 2e-60))
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = Float64(t_1 / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) * ky;
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = t_1 * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	elseif (sin(kx) <= 2.5e-119)
		tmp = sin(th);
	elseif ((sin(kx) <= 1e-98) || ~((sin(kx) <= 2e-60)))
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = t_1 / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(t$95$1 * N[Sqrt[N[(1.0 / N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2.5e-119], N[Sin[th], $MachinePrecision], If[Or[LessEqual[N[Sin[kx], $MachinePrecision], 1e-98], N[Not[LessEqual[N[Sin[kx], $MachinePrecision], 2e-60]], $MachinePrecision]], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / ky), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 98.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 64.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

   8. Taylor expanded in ky around 0 57.7%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   if -0.0050000000000000001 < (sin.f64 kx) < 2.49999999999999996e-119

   1. Initial program 82.0%

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

      1. unpow2 82.0%

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

      2. sqr-neg 82.0%

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

      3. sin-neg 82.0%

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

      4. sin-neg 82.0%

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

      5. unpow2 82.0%

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

      6. associate-*l/ 79.1%

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

      7. associate-/l* 81.9%

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

      8. unpow2 81.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 41.9%

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

   if 2.49999999999999996e-119 < (sin.f64 kx) < 9.99999999999999939e-99 or 1.9999999999999999e-60 < (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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-undefine 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0 69.7%

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

   if 9.99999999999999939e-99 < (sin.f64 kx) < 1.9999999999999999e-60

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 23.3%

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

      1. associate-*r/ 43.1%

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

      2. sin-mult 24.4%

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

      3. associate-/l/ 24.4%

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

      4. +-commutative 24.4%

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

   7. Applied egg-rr 24.4%

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

   8. Taylor expanded in ky around 0 43.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 43.1%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 43.1%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 43.1%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 43.1%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2.5 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-98} \lor \neg \left(\sin kx \leq 2 \cdot 10^{-60}\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2.5 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-98} \lor \neg \left(\sin kx \leq 2 \cdot 10^{-60}\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (* (sin th) (/ ky (sqrt (- 0.5 (* 0.5 (cos (* kx 2.0)))))))
   (if (<= (sin kx) 2.5e-119)
     (sin th)
     (if (or (<= (sin kx) 1e-98) (not (<= (sin kx) 2e-60)))
       (* (sin th) (/ (sin ky) (sin kx)))
       (/ (* (sin th) ky) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = sin(th) * (ky / sqrt((0.5 - (0.5 * cos((kx * 2.0))))));
	} else if (sin(kx) <= 2.5e-119) {
		tmp = sin(th);
	} else if ((sin(kx) <= 1e-98) || !(sin(kx) <= 2e-60)) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = (sin(th) * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = sin(th) * (ky / sqrt((0.5d0 - (0.5d0 * cos((kx * 2.0d0))))))
    else if (sin(kx) <= 2.5d-119) then
        tmp = sin(th)
    else if ((sin(kx) <= 1d-98) .or. (.not. (sin(kx) <= 2d-60))) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else
        tmp = (sin(th) * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = Math.sin(th) * (ky / Math.sqrt((0.5 - (0.5 * Math.cos((kx * 2.0))))));
	} else if (Math.sin(kx) <= 2.5e-119) {
		tmp = Math.sin(th);
	} else if ((Math.sin(kx) <= 1e-98) || !(Math.sin(kx) <= 2e-60)) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = (Math.sin(th) * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = math.sin(th) * (ky / math.sqrt((0.5 - (0.5 * math.cos((kx * 2.0))))))
	elif math.sin(kx) <= 2.5e-119:
		tmp = math.sin(th)
	elif (math.sin(kx) <= 1e-98) or not (math.sin(kx) <= 2e-60):
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = (math.sin(th) * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(sin(th) * Float64(ky / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0)))))));
	elseif (sin(kx) <= 2.5e-119)
		tmp = sin(th);
	elseif ((sin(kx) <= 1e-98) || !(sin(kx) <= 2e-60))
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = Float64(Float64(sin(th) * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = sin(th) * (ky / sqrt((0.5 - (0.5 * cos((kx * 2.0))))));
	elseif (sin(kx) <= 2.5e-119)
		tmp = sin(th);
	elseif ((sin(kx) <= 1e-98) || ~((sin(kx) <= 2e-60)))
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = (sin(th) * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2.5e-119], N[Sin[th], $MachinePrecision], If[Or[LessEqual[N[Sin[kx], $MachinePrecision], 1e-98], N[Not[LessEqual[N[Sin[kx], $MachinePrecision], 2e-60]], $MachinePrecision]], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / ky), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 98.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 64.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

   8. Taylor expanded in ky around 0 57.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   if -0.0050000000000000001 < (sin.f64 kx) < 2.49999999999999996e-119

   1. Initial program 82.0%

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

      1. unpow2 82.0%

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

      2. sqr-neg 82.0%

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

      3. sin-neg 82.0%

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

      4. sin-neg 82.0%

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

      5. unpow2 82.0%

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

      6. associate-*l/ 79.1%

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

      7. associate-/l* 81.9%

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

      8. unpow2 81.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in kx around 0 41.9%

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

   if 2.49999999999999996e-119 < (sin.f64 kx) < 9.99999999999999939e-99 or 1.9999999999999999e-60 < (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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. hypot-undefine 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0 69.7%

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

   if 9.99999999999999939e-99 < (sin.f64 kx) < 1.9999999999999999e-60

   1. Initial program 100.0%

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

      1. unpow2 100.0%

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

      2. sqr-neg 100.0%

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

      3. sin-neg 100.0%

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

      4. sin-neg 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l/ 81.8%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 23.3%

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

      1. associate-*r/ 43.1%

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

      2. sin-mult 24.4%

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

      3. associate-/l/ 24.4%

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

      4. +-commutative 24.4%

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

   7. Applied egg-rr 24.4%

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

   8. Taylor expanded in ky around 0 43.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 43.1%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 43.1%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 43.1%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 43.1%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 43.1%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2.5 \cdot 10^{-119}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-98} \lor \neg \left(\sin kx \leq 2 \cdot 10^{-60}\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (* (* (sin th) ky) (sqrt (/ 1.0 (- 0.5 (* 0.5 (cos (* kx 2.0)))))))
   (if (<= (sin kx) 2e-73)
     (* (sin ky) (/ (sin th) (fabs (sin ky))))
     (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (sin(th) * ky) * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	} else if (sin(kx) <= 2e-73) {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	} 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) <= (-0.005d0)) then
        tmp = (sin(th) * ky) * sqrt((1.0d0 / (0.5d0 - (0.5d0 * cos((kx * 2.0d0))))))
    else if (sin(kx) <= 2d-73) then
        tmp = sin(ky) * (sin(th) / abs(sin(ky)))
    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) <= -0.005) {
		tmp = (Math.sin(th) * ky) * Math.sqrt((1.0 / (0.5 - (0.5 * Math.cos((kx * 2.0))))));
	} else if (Math.sin(kx) <= 2e-73) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	} 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) <= -0.005:
		tmp = (math.sin(th) * ky) * math.sqrt((1.0 / (0.5 - (0.5 * math.cos((kx * 2.0))))))
	elif math.sin(kx) <= 2e-73:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	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) <= -0.005)
		tmp = Float64(Float64(sin(th) * ky) * sqrt(Float64(1.0 / Float64(0.5 - Float64(0.5 * cos(Float64(kx * 2.0)))))));
	elseif (sin(kx) <= 2e-73)
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	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) <= -0.005)
		tmp = (sin(th) * ky) * sqrt((1.0 / (0.5 - (0.5 * cos((kx * 2.0))))));
	elseif (sin(kx) <= 2e-73)
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 - N[(0.5 * N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-73], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $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) < -0.0050000000000000001

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 98.0%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 98.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 64.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

   8. Taylor expanded in ky around 0 57.7%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   if -0.0050000000000000001 < (sin.f64 kx) < 1.99999999999999999e-73

   1. Initial program 83.7%

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

      1. unpow2 83.7%

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

      2. sqr-neg 83.7%

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

      3. sin-neg 83.7%

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

      4. sin-neg 83.7%

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

      5. unpow2 83.7%

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

      6. associate-*l/ 77.9%

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

      7. associate-/l* 83.5%

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

      8. unpow2 83.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 41.5%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-prod 67.2%

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

      3. rem-sqrt-square 75.0%

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

   7. Applied egg-rr 75.0%

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

   if 1.99999999999999999e-73 < (sin.f64 kx)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

      3. unpow2 99.4%

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

      4. hypot-undefine 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0 71.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{0.5 - 0.5 \cdot \cos \left(kx \cdot 2\right)}}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. +-commutative 92.3%

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

   2. unpow2 92.3%

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

   3. unpow2 92.3%

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

   4. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 6: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. unpow2 92.3%

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

   2. sqr-neg 92.3%

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

   3. sin-neg 92.3%

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

   4. sin-neg 92.3%

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

   5. unpow2 92.3%

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

   6. associate-*l/ 89.6%

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

   7. associate-/l* 92.2%

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

   8. unpow2 92.2%

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

3. Simplified 99.6%

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

Alternative 7: 65.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0035:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;ky \leq 1.3 \cdot 10^{+38} \lor \neg \left(ky \leq 2.2 \cdot 10^{+148}\right):\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0035)
   (/ (sin th) (/ (hypot ky (sin kx)) (sin ky)))
   (if (or (<= ky 1.3e+38) (not (<= ky 2.2e+148)))
     (* (sin ky) (/ (sin th) (fabs (sin ky))))
     (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0035) {
		tmp = sin(th) / (hypot(ky, sin(kx)) / sin(ky));
	} else if ((ky <= 1.3e+38) || !(ky <= 2.2e+148)) {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0035) {
		tmp = Math.sin(th) / (Math.hypot(ky, Math.sin(kx)) / Math.sin(ky));
	} else if ((ky <= 1.3e+38) || !(ky <= 2.2e+148)) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0035:
		tmp = math.sin(th) / (math.hypot(ky, math.sin(kx)) / math.sin(ky))
	elif (ky <= 1.3e+38) or not (ky <= 2.2e+148):
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0035)
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / sin(ky)));
	elseif ((ky <= 1.3e+38) || !(ky <= 2.2e+148))
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0035)
		tmp = sin(th) / (hypot(ky, sin(kx)) / sin(ky));
	elseif ((ky <= 1.3e+38) || ~((ky <= 2.2e+148)))
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	else
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0035], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[ky, 1.3e+38], N[Not[LessEqual[ky, 2.2e+148]], $MachinePrecision]], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.00350000000000000007

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.8%

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

      7. associate-/l* 90.1%

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

      8. unpow2 90.1%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 92.3%

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

      2. hypot-undefine 86.8%

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

      3. unpow2 86.8%

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

      4. unpow2 86.8%

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

      5. +-commutative 86.8%

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

      6. associate-*l/ 90.2%

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

      7. *-commutative 90.2%

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

      8. clear-num 90.2%

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

      9. un-div-inv 90.2%

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

      10. +-commutative 90.2%

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

      11. unpow2 90.2%

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

      12. unpow2 90.2%

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

      13. hypot-undefine 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in ky around 0 78.1%

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

   if 0.00350000000000000007 < ky < 1.3e38 or 2.1999999999999999e148 < ky

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 43.5%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-prod 68.4%

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

      3. rem-sqrt-square 68.4%

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

   7. Applied egg-rr 68.4%

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

   if 1.3e38 < ky < 2.1999999999999999e148

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in th around 0 65.2%

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

      1. associate-*l/ 65.1%

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

      2. +-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. hypot-undefine 65.1%

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

      6. *-lft-identity 65.1%

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

      7. hypot-undefine 65.1%

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

      8. unpow2 65.1%

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

      9. unpow2 65.1%

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

      10. +-commutative 65.1%

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

      11. unpow2 65.1%

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

      12. unpow2 65.1%

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

      13. hypot-define 65.1%

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

   9. Simplified 65.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0035:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{\sin ky}}\\ \mathbf{elif}\;ky \leq 1.3 \cdot 10^{+38} \lor \neg \left(ky \leq 2.2 \cdot 10^{+148}\right):\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0115:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 9.2 \cdot 10^{+43} \lor \neg \left(ky \leq 7.7 \cdot 10^{+146}\right):\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0115)
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
   (if (or (<= ky 9.2e+43) (not (<= ky 7.7e+146)))
     (* (sin ky) (/ (sin th) (fabs (sin ky))))
     (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0115) {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	} else if ((ky <= 9.2e+43) || !(ky <= 7.7e+146)) {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	} else {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0115) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	} else if ((ky <= 9.2e+43) || !(ky <= 7.7e+146)) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	} else {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0115:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	elif (ky <= 9.2e+43) or not (ky <= 7.7e+146):
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	else:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0115)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	elseif ((ky <= 9.2e+43) || !(ky <= 7.7e+146))
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	else
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0115)
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	elseif ((ky <= 9.2e+43) || ~((ky <= 7.7e+146)))
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	else
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0115], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[ky, 9.2e+43], N[Not[LessEqual[ky, 7.7e+146]], $MachinePrecision]], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 0.0115

   1. Initial program 90.2%

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

      1. +-commutative 90.2%

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

      2. unpow2 90.2%

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

      3. unpow2 90.2%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0 78.1%

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

   if 0.0115 < ky < 9.200000000000001e43 or 7.7000000000000002e146 < ky

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 43.5%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-prod 68.4%

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

      3. rem-sqrt-square 68.4%

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

   7. Applied egg-rr 68.4%

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

   if 9.200000000000001e43 < ky < 7.7000000000000002e146

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in th around 0 65.2%

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

      1. associate-*l/ 65.1%

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

      2. +-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. hypot-undefine 65.1%

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

      6. *-lft-identity 65.1%

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

      7. hypot-undefine 65.1%

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

      8. unpow2 65.1%

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

      9. unpow2 65.1%

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

      10. +-commutative 65.1%

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

      11. unpow2 65.1%

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

      12. unpow2 65.1%

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

      13. hypot-define 65.1%

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

   9. Simplified 65.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0115:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 9.2 \cdot 10^{+43} \lor \neg \left(ky \leq 7.7 \cdot 10^{+146}\right):\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (sqrt (pow (sin th) 2.0))
   (if (<= (sin ky) 1e-104)
     (* (sin th) (/ ky (sin kx)))
     (/ (* (sin th) ky) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = sqrt(pow(sin(th), 2.0));
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = (sin(th) * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.01d0)) then
        tmp = sqrt((sin(th) ** 2.0d0))
    else if (sin(ky) <= 1d-104) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = (sin(th) * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.sqrt(Math.pow(Math.sin(th), 2.0));
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = (Math.sin(th) * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.sqrt(math.pow(math.sin(th), 2.0))
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = (math.sin(th) * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = Float64(Float64(sin(th) * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = sqrt((sin(th) ^ 2.0));
	elseif (sin(ky) <= 1e-104)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = (sin(th) * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / ky), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 21.8%

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

      3. pow2 21.8%

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

      4. associate-*r/ 21.8%

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

      5. *-commutative 21.8%

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

      6. associate-/l* 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. *-inverses 21.8%

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

      2. *-commutative 21.8%

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

   9. Simplified 21.8%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 85.8%

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

      1. +-commutative 85.8%

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

      2. unpow2 85.8%

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

      3. unpow2 85.8%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0 45.8%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 60.7%

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

      1. associate-*r/ 61.9%

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

      2. sin-mult 6.7%

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

      3. associate-/l/ 6.7%

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

      4. +-commutative 6.7%

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

   7. Applied egg-rr 6.7%

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

   8. Taylor expanded in ky around 0 61.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 61.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 61.9%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 61.9%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (fabs (sin th))
   (if (<= (sin ky) 1e-104)
     (* (sin th) (/ ky (sin kx)))
     (/ (* (sin th) ky) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-104) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = (sin(th) * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.01d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-104) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = (sin(th) * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = (Math.sin(th) * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-104:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = (math.sin(th) * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = Float64(Float64(sin(th) * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = (sin(th) * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / ky), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 21.8%

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

      3. pow2 21.8%

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

      4. associate-*r/ 21.8%

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

      5. *-commutative 21.8%

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

      6. associate-/l* 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. unpow2 21.8%

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

      2. rem-sqrt-square 24.6%

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

      3. *-inverses 24.6%

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

      4. *-rgt-identity 24.6%

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

   9. Simplified 24.6%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 85.8%

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

      1. +-commutative 85.8%

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

      2. unpow2 85.8%

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

      3. unpow2 85.8%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0 45.8%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 60.7%

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

      1. associate-*r/ 61.9%

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

      2. sin-mult 6.7%

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

      3. associate-/l/ 6.7%

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

      4. +-commutative 6.7%

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

   7. Applied egg-rr 6.7%

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

   8. Taylor expanded in ky around 0 61.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 61.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 61.9%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 61.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (fabs (sin th))
   (if (<= (sin ky) 1e-104)
     (* ky (/ (sin th) (sin kx)))
     (/ (* (sin th) ky) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 1e-104) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = (sin(th) * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.01d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 1d-104) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = (sin(th) * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 1e-104) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = (Math.sin(th) * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.01:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 1e-104:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = (math.sin(th) * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = Float64(Float64(sin(th) * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.01)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 1e-104)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = (sin(th) * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-104], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / ky), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 2.5%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 21.8%

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

      3. pow2 21.8%

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

      4. associate-*r/ 21.8%

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

      5. *-commutative 21.8%

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

      6. associate-/l* 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. unpow2 21.8%

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

      2. rem-sqrt-square 24.6%

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

      3. *-inverses 24.6%

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

      4. *-rgt-identity 24.6%

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

   9. Simplified 24.6%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 9.99999999999999927e-105

   1. Initial program 85.8%

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

      1. unpow2 85.8%

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

      2. sqr-neg 85.8%

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

      3. sin-neg 85.8%

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

      4. sin-neg 85.8%

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

      5. unpow2 85.8%

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

      6. associate-*l/ 80.9%

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

      7. associate-/l* 85.8%

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

      8. unpow2 85.8%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 42.2%

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

      1. associate-/l* 45.9%

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

   7. Simplified 45.9%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 60.7%

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

      1. associate-*r/ 61.9%

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

      2. sin-mult 6.7%

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

      3. associate-/l/ 6.7%

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

      4. +-commutative 6.7%

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

   7. Applied egg-rr 6.7%

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

   8. Taylor expanded in ky around 0 61.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 61.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 61.9%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 61.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-104}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.007:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.007)
   (* (sin th) (/ (sin ky) (hypot ky (sin kx))))
   (* (sin ky) (/ (sin th) (fabs (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.007) {
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.007) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.007:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.007)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.007)
		tmp = sin(th) * (sin(ky) / hypot(ky, sin(kx)));
	else
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.007], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00700000000000000015

   1. Initial program 90.2%

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

      1. +-commutative 90.2%

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

      2. unpow2 90.2%

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

      3. unpow2 90.2%

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

      4. hypot-undefine 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ky around 0 78.1%

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

   if 0.00700000000000000015 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 37.7%

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

      1. add-sqr-sqrt 36.4%

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

      2. sqrt-prod 58.3%

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

      3. rem-sqrt-square 58.3%

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

   7. Applied egg-rr 58.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.007:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0125:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0125)
   (* (sin ky) (/ (sin th) (hypot ky (sin kx))))
   (* (sin ky) (/ (sin th) (fabs (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0125) {
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0125) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.0125:
		tmp = math.sin(ky) * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0125)
		tmp = Float64(sin(ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.0125)
		tmp = sin(ky) * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0125], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.012500000000000001

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.8%

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

      7. associate-/l* 90.1%

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

      8. unpow2 90.1%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 78.1%

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

   if 0.012500000000000001 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.6%

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

      8. unpow2 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 37.7%

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

      1. add-sqr-sqrt 36.4%

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

      2. sqrt-prod 58.3%

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

      3. rem-sqrt-square 58.3%

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

   7. Applied egg-rr 58.3%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\color{blue}{\left|\sin ky\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.9 \cdot 10^{-73}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.9e-73)
   (* (sin ky) (/ (sin th) (fabs (sin ky))))
   (* (sin th) (/ (sin ky) (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.9e-73) {
		tmp = sin(ky) * (sin(th) / fabs(sin(ky)));
	} else {
		tmp = sin(th) * (sin(ky) / fabs(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 (kx <= 2.9d-73) then
        tmp = sin(ky) * (sin(th) / abs(sin(ky)))
    else
        tmp = sin(th) * (sin(ky) / abs(sin(kx)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.9e-73) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(ky)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 2.9e-73:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(ky)))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.9e-73)
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 2.9e-73)
		tmp = sin(ky) * (sin(th) / abs(sin(ky)));
	else
		tmp = sin(th) * (sin(ky) / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.9e-73], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.9e-73

   1. Initial program 89.8%

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

      1. unpow2 89.8%

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

      2. sqr-neg 89.8%

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

      3. sin-neg 89.8%

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

      4. sin-neg 89.8%

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

      5. unpow2 89.8%

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

      6. associate-*l/ 86.2%

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

      7. associate-/l* 89.7%

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

      8. unpow2 89.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.8%

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

      1. add-sqr-sqrt 26.2%

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

      2. sqrt-prod 45.2%

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

      3. rem-sqrt-square 50.2%

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

   7. Applied egg-rr 50.2%

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

   if 2.9e-73 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. div-sub 90.3%

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

      2. +-inverses 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      3. cos-0 90.3%

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

      4. metadata-eval 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{0.5} - \frac{\cos \left(kx + kx\right)}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      5. count-2 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot kx\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      6. *-commutative 90.3%

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \frac{\cos \color{blue}{\left(kx \cdot 2\right)}}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   6. Simplified 90.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \frac{\cos \left(kx \cdot 2\right)}{2}\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0 56.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 56.8%

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)} \cdot \sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}}} \cdot \sin th \]

      2. rem-sqrt-square 56.8%

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}\right|}} \cdot \sin th \]

      3. sqr-sin-a 66.2%

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

      4. sqrt-unprod 35.5%

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

      5. add-sqr-sqrt 66.2%

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

   9. Applied egg-rr 66.2%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.9 \cdot 10^{-73}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1e-104) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = (sin(th) * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 1d-104) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else
        tmp = (sin(th) * ky) / ky
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. unpow2 89.4%

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

      3. unpow2 89.4%

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

      4. hypot-undefine 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in ky around 0 36.2%

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

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.7%

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

      8. unpow2 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 60.7%

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

      1. associate-*r/ 61.9%

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

      2. sin-mult 6.7%

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

      3. associate-/l/ 6.7%

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

      4. +-commutative 6.7%

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

   7. Applied egg-rr 6.7%

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

   8. Taylor expanded in ky around 0 61.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 61.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 61.9%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 61.9%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 61.9%

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

4. Final simplification 43.4%

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

Alternative 16: 27.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.7 \cdot 10^{-101}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.7e-101) (sin th) (/ (* (sin th) ky) ky)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.7e-101) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 4.7d-101) then
        tmp = sin(th)
    else
        tmp = (sin(th) * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.7e-101) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.7e-101:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.7e-101)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.7e-101)
		tmp = sin(th);
	else
		tmp = (sin(th) * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.7e-101], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / ky), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 4.6999999999999999e-101

   1. Initial program 89.6%

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

      1. unpow2 89.6%

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

      2. sqr-neg 89.6%

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

      3. sin-neg 89.6%

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

      4. sin-neg 89.6%

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

      5. unpow2 89.6%

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

      6. associate-*l/ 86.4%

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

      7. associate-/l* 89.5%

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

      8. unpow2 89.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.8%

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

   if 4.6999999999999999e-101 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 98.1%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 10.9%

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

      1. associate-*r/ 23.0%

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

      2. sin-mult 18.6%

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

      3. associate-/l/ 18.6%

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

      4. +-commutative 18.6%

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

   7. Applied egg-rr 18.6%

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

   8. Taylor expanded in ky around 0 23.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 23.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 23.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 23.0%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 23.0%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in ky around 0 23.0%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.7 \cdot 10^{-101}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin th + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.5e-5) (sin th) (+ (+ (sin th) 1.0) -1.0)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.5e-5) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) + 1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 6.5d-5) then
        tmp = sin(th)
    else
        tmp = (sin(th) + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.5e-5) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 6.5e-5:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) + 1.0) + -1.0
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.5e-5)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 6.5e-5)
		tmp = sin(th);
	else
		tmp = (sin(th) + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.5e-5], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.49999999999999943e-5

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.7%

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

      7. associate-/l* 90.1%

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

      8. unpow2 90.1%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.5%

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

   if 6.49999999999999943e-5 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 8.9%

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

      1. expm1-log1p-u 8.9%

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

      2. expm1-undefine 22.5%

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

      3. associate-*r/ 22.5%

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

      4. *-commutative 22.5%

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

      5. associate-/l* 22.5%

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

   7. Applied egg-rr 22.5%

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

      1. expm1-define 8.9%

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

      2. *-inverses 8.9%

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

      3. *-rgt-identity 8.9%

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

   9. Simplified 8.9%

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

       1. expm1-undefine 22.5%

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

       2. log1p-undefine 22.5%

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

       3. rem-exp-log 22.5%

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

       4. +-commutative 22.5%

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

   11. Applied egg-rr 22.5%

       \[\leadsto \color{blue}{\left(\sin th + 1\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin th + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 980:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\log \left(th + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 980.0) (sin th) (log (+ th 1.0))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 980.0) {
		tmp = sin(th);
	} else {
		tmp = log((th + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 980.0d0) then
        tmp = sin(th)
    else
        tmp = log((th + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 980.0) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.log((th + 1.0));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 980.0:
		tmp = math.sin(th)
	else:
		tmp = math.log((th + 1.0))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 980.0)
		tmp = sin(th);
	else
		tmp = log(Float64(th + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 980.0)
		tmp = sin(th);
	else
		tmp = log((th + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 980.0], N[Sin[th], $MachinePrecision], N[Log[N[(th + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 980

   1. Initial program 90.2%

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

      1. unpow2 90.2%

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

      2. sqr-neg 90.2%

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

      3. sin-neg 90.2%

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

      4. sin-neg 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 86.8%

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

      7. associate-/l* 90.1%

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

      8. unpow2 90.1%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.6%

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

   if 980 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 8.2%

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

      1. add-log-exp 22.0%

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

      2. associate-*r/ 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-/l* 22.0%

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

   7. Applied egg-rr 22.0%

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

   8. Taylor expanded in th around 0 19.3%

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

      1. +-commutative 19.3%

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

   10. Simplified 19.3%

       \[\leadsto \log \color{blue}{\left(th + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 25.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.1 \cdot 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.1e-67) (sin th) (/ (* th ky) ky)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.1e-67) {
		tmp = sin(th);
	} else {
		tmp = (th * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 2.1d-67) then
        tmp = sin(th)
    else
        tmp = (th * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.1e-67) {
		tmp = Math.sin(th);
	} else {
		tmp = (th * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 2.1e-67:
		tmp = math.sin(th)
	else:
		tmp = (th * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.1e-67)
		tmp = sin(th);
	else
		tmp = Float64(Float64(th * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 2.1e-67)
		tmp = sin(th);
	else
		tmp = (th * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.1e-67], N[Sin[th], $MachinePrecision], N[(N[(th * ky), $MachinePrecision] / ky), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.1000000000000002e-67

   1. Initial program 89.8%

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

      1. unpow2 89.8%

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

      2. sqr-neg 89.8%

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

      3. sin-neg 89.8%

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

      4. sin-neg 89.8%

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

      5. unpow2 89.8%

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

      6. associate-*l/ 86.2%

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

      7. associate-/l* 89.7%

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

      8. unpow2 89.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.8%

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

   if 2.1000000000000002e-67 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 99.4%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 9.9%

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

      1. associate-*r/ 22.7%

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

      2. sin-mult 19.4%

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

      3. associate-/l/ 19.4%

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

      4. +-commutative 19.4%

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

   7. Applied egg-rr 19.4%

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

   8. Taylor expanded in ky around 0 22.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 22.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 22.7%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 22.7%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 22.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in th around 0 19.5%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.1 \cdot 10^{-67}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 16.6% accurate, 70.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 10^{-86}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot ky}{ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (if (<= kx 1e-86) th (/ (* th ky) ky)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1e-86) {
		tmp = th;
	} else {
		tmp = (th * ky) / ky;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1d-86) then
        tmp = th
    else
        tmp = (th * ky) / ky
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1e-86) {
		tmp = th;
	} else {
		tmp = (th * ky) / ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1e-86:
		tmp = th
	else:
		tmp = (th * ky) / ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1e-86)
		tmp = th;
	else
		tmp = Float64(Float64(th * ky) / ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1e-86)
		tmp = th;
	else
		tmp = (th * ky) / ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1e-86], th, N[(N[(th * ky), $MachinePrecision] / ky), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.00000000000000008e-86

   1. Initial program 89.7%

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

      1. unpow2 89.7%

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

      2. sqr-neg 89.7%

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

      3. sin-neg 89.7%

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

      4. sin-neg 89.7%

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

      5. unpow2 89.7%

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

      6. associate-*l/ 86.6%

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

      7. associate-/l* 89.6%

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

      8. unpow2 89.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 28.0%

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

   6. Taylor expanded in th around 0 19.0%

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

   if 1.00000000000000008e-86 < kx

   1. Initial program 99.5%

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

      1. unpow2 99.5%

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

      2. sqr-neg 99.5%

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

      3. sin-neg 99.5%

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

      4. sin-neg 99.5%

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

      5. unpow2 99.5%

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

      6. associate-*l/ 98.1%

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

      7. associate-/l* 99.5%

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

      8. unpow2 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 9.7%

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

      1. associate-*r/ 22.2%

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

      2. sin-mult 19.0%

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

      3. associate-/l/ 19.0%

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

      4. +-commutative 19.0%

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

   7. Applied egg-rr 19.0%

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

   8. Taylor expanded in ky around 0 22.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right)\right) + 0.5 \cdot \left(\cos \left(-th\right) - \cos th\right)}{ky}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 22.2%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(ky \cdot \left(\sin th - \sin \left(-th\right)\right) + \left(\cos \left(-th\right) - \cos th\right)\right)}}{ky} \]

      2. fma-define 22.2%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(ky, \sin th - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}}{ky} \]

      3. *-lft-identity 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{1 \cdot \sin th} - \sin \left(-th\right), \cos \left(-th\right) - \cos th\right)}{ky} \]

      4. sin-neg 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{\left(-\sin th\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      5. neg-mul-1 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, 1 \cdot \sin th - \color{blue}{-1 \cdot \sin th}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      6. distribute-rgt-out-- 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \color{blue}{\sin th \cdot \left(1 - -1\right)}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      7. metadata-eval 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot \color{blue}{2}, \cos \left(-th\right) - \cos th\right)}{ky} \]

      8. cos-neg 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{\cos th} - \cos th\right)}{ky} \]

      9. +-inverses 22.2%

         \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, \color{blue}{0}\right)}{ky} \]

   10. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(ky, \sin th \cdot 2, 0\right)}{ky}} \]

   11. Taylor expanded in th around 0 19.1%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 10^{-86}:\\ \;\;\;\;th\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot ky}{ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 13.7% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.3%

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

   1. unpow2 92.3%

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

   2. sqr-neg 92.3%

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

   3. sin-neg 92.3%

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

   4. sin-neg 92.3%

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

   5. unpow2 92.3%

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

   6. associate-*l/ 89.6%

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

   7. associate-/l* 92.2%

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

   8. unpow2 92.2%

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

3. Simplified 99.6%

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

5. Taylor expanded in kx around 0 23.2%

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

6. Taylor expanded in th around 0 15.8%

   \[\leadsto \color{blue}{th} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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)))