Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 19.9s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. +-commutative 93.7%

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

   2. unpow2 93.7%

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

   3. unpow2 93.7%

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

   4. hypot-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ t_2 := \sin ky \cdot \left|t_1\right|\\ \mathbf{if}\;\sin kx \leq -0.82:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin kx \leq -0.505:\\ \;\;\;\;\left|\sin ky \cdot t_1\right|\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))) (t_2 (* (sin ky) (fabs t_1))))
   (if (<= (sin kx) -0.82)
     t_2
     (if (<= (sin kx) -0.505)
       (fabs (* (sin ky) t_1))
       (if (<= (sin kx) -0.02)
         t_2
         (if (<= (sin kx) 2e-19)
           (sin th)
           (* (sin th) (/ (sin ky) (sin kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double t_2 = sin(ky) * fabs(t_1);
	double tmp;
	if (sin(kx) <= -0.82) {
		tmp = t_2;
	} else if (sin(kx) <= -0.505) {
		tmp = fabs((sin(ky) * t_1));
	} else if (sin(kx) <= -0.02) {
		tmp = t_2;
	} else if (sin(kx) <= 2e-19) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    t_2 = sin(ky) * abs(t_1)
    if (sin(kx) <= (-0.82d0)) then
        tmp = t_2
    else if (sin(kx) <= (-0.505d0)) then
        tmp = abs((sin(ky) * t_1))
    else if (sin(kx) <= (-0.02d0)) then
        tmp = t_2
    else if (sin(kx) <= 2d-19) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double t_2 = Math.sin(ky) * Math.abs(t_1);
	double tmp;
	if (Math.sin(kx) <= -0.82) {
		tmp = t_2;
	} else if (Math.sin(kx) <= -0.505) {
		tmp = Math.abs((Math.sin(ky) * t_1));
	} else if (Math.sin(kx) <= -0.02) {
		tmp = t_2;
	} else if (Math.sin(kx) <= 2e-19) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	t_2 = math.sin(ky) * math.fabs(t_1)
	tmp = 0
	if math.sin(kx) <= -0.82:
		tmp = t_2
	elif math.sin(kx) <= -0.505:
		tmp = math.fabs((math.sin(ky) * t_1))
	elif math.sin(kx) <= -0.02:
		tmp = t_2
	elif math.sin(kx) <= 2e-19:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	t_2 = Float64(sin(ky) * abs(t_1))
	tmp = 0.0
	if (sin(kx) <= -0.82)
		tmp = t_2;
	elseif (sin(kx) <= -0.505)
		tmp = abs(Float64(sin(ky) * t_1));
	elseif (sin(kx) <= -0.02)
		tmp = t_2;
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	t_2 = sin(ky) * abs(t_1);
	tmp = 0.0;
	if (sin(kx) <= -0.82)
		tmp = t_2;
	elseif (sin(kx) <= -0.505)
		tmp = abs((sin(ky) * t_1));
	elseif (sin(kx) <= -0.02)
		tmp = t_2;
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.82], t$95$2, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.505], N[Abs[N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -0.02], t$95$2, If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-19], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.819999999999999951 or -0.505000000000000004 < (sin.f64 kx) < -0.0200000000000000004

   1. Initial program 99.5%

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

      1. associate-*l/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.4%

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

      4. +-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in ky around 0 9.0%

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

      1. add-sqr-sqrt 5.8%

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

      2. sqrt-unprod 40.2%

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

      3. pow2 40.2%

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

   6. Applied egg-rr 40.2%

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

      1. unpow2 40.2%

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

      2. rem-sqrt-square 40.2%

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

   8. Simplified 40.2%

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

   if -0.819999999999999951 < (sin.f64 kx) < -0.505000000000000004

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0 13.5%

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

      1. associate-*l/ 13.5%

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

      2. *-commutative 13.5%

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

      3. add-sqr-sqrt 7.3%

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

      4. sqrt-unprod 41.4%

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

      5. pow2 41.4%

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

      6. associate-/l* 41.5%

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

      7. associate-/r/ 41.4%

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

   4. Applied egg-rr 41.4%

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

      1. unpow2 41.4%

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

      2. rem-sqrt-square 65.4%

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

      3. *-commutative 65.4%

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

   6. Simplified 65.4%

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

   if -0.0200000000000000004 < (sin.f64 kx) < 2e-19

   1. Initial program 87.1%

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

      1. +-commutative 87.1%

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

      2. unpow2 87.1%

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

      3. unpow2 87.1%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 43.9%

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

   if 2e-19 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 52.7%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.82:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -0.505:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -0.02:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 3: 44.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.82:\\ \;\;\;\;\sin ky \cdot \left|t_1\right|\\ \mathbf{elif}\;\sin kx \leq -0.5:\\ \;\;\;\;\left|\sin ky \cdot t_1\right|\\ \mathbf{elif}\;\sin kx \leq -0.005:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.82)
     (* (sin ky) (fabs t_1))
     (if (<= (sin kx) -0.5)
       (fabs (* (sin ky) t_1))
       (if (<= (sin kx) -0.005)
         (* (sin th) (/ (fabs (sin ky)) (sin kx)))
         (if (<= (sin kx) 2e-19)
           (sin th)
           (* (sin th) (/ (sin ky) (sin kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.82) {
		tmp = sin(ky) * fabs(t_1);
	} else if (sin(kx) <= -0.5) {
		tmp = fabs((sin(ky) * t_1));
	} else if (sin(kx) <= -0.005) {
		tmp = sin(th) * (fabs(sin(ky)) / sin(kx));
	} else if (sin(kx) <= 2e-19) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    if (sin(kx) <= (-0.82d0)) then
        tmp = sin(ky) * abs(t_1)
    else if (sin(kx) <= (-0.5d0)) then
        tmp = abs((sin(ky) * t_1))
    else if (sin(kx) <= (-0.005d0)) then
        tmp = sin(th) * (abs(sin(ky)) / sin(kx))
    else if (sin(kx) <= 2d-19) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.82) {
		tmp = Math.sin(ky) * Math.abs(t_1);
	} else if (Math.sin(kx) <= -0.5) {
		tmp = Math.abs((Math.sin(ky) * t_1));
	} else if (Math.sin(kx) <= -0.005) {
		tmp = Math.sin(th) * (Math.abs(Math.sin(ky)) / Math.sin(kx));
	} else if (Math.sin(kx) <= 2e-19) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.82:
		tmp = math.sin(ky) * math.fabs(t_1)
	elif math.sin(kx) <= -0.5:
		tmp = math.fabs((math.sin(ky) * t_1))
	elif math.sin(kx) <= -0.005:
		tmp = math.sin(th) * (math.fabs(math.sin(ky)) / math.sin(kx))
	elif math.sin(kx) <= 2e-19:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.82)
		tmp = Float64(sin(ky) * abs(t_1));
	elseif (sin(kx) <= -0.5)
		tmp = abs(Float64(sin(ky) * t_1));
	elseif (sin(kx) <= -0.005)
		tmp = Float64(sin(th) * Float64(abs(sin(ky)) / sin(kx)));
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.82)
		tmp = sin(ky) * abs(t_1);
	elseif (sin(kx) <= -0.5)
		tmp = abs((sin(ky) * t_1));
	elseif (sin(kx) <= -0.005)
		tmp = sin(th) * (abs(sin(ky)) / sin(kx));
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.82], N[(N[Sin[ky], $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -0.5], N[Abs[N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-19], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 kx) < -0.819999999999999951

   1. Initial program 99.5%

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

      1. associate-*l/ 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l/ 99.3%

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

      4. +-commutative 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in ky around 0 2.3%

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

      1. add-sqr-sqrt 0.7%

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

      2. sqrt-unprod 40.3%

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

      3. pow2 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. unpow2 40.3%

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

      2. rem-sqrt-square 40.2%

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

   8. Simplified 40.2%

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

   if -0.819999999999999951 < (sin.f64 kx) < -0.5

   1. Initial program 99.3%

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

   2. Taylor expanded in ky around 0 12.8%

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

      1. associate-*l/ 12.8%

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

      2. *-commutative 12.8%

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

      3. add-sqr-sqrt 6.8%

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

      4. sqrt-unprod 44.9%

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

      5. pow2 44.9%

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

      6. associate-/l* 45.1%

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

      7. associate-/r/ 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. unpow2 44.9%

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

      2. rem-sqrt-square 67.5%

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

      3. *-commutative 67.5%

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

   6. Simplified 67.5%

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

   if -0.5 < (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 15.9%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-prod 24.0%

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

      3. rem-sqrt-square 38.7%

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

   4. Applied egg-rr 38.7%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 2e-19

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. unpow2 86.8%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 44.6%

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

   if 2e-19 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 52.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.82:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -0.5:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -0.005:\\ \;\;\;\;\sin th \cdot \frac{\left|\sin ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 4: 44.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -0.82:\\ \;\;\;\;\sin ky \cdot \left|t_1\right|\\ \mathbf{elif}\;\sin kx \leq -0.5:\\ \;\;\;\;\left|\sin ky \cdot t_1\right|\\ \mathbf{elif}\;\sin kx \leq -0.005:\\ \;\;\;\;t_1 \cdot \left|\sin ky\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (sin kx))))
   (if (<= (sin kx) -0.82)
     (* (sin ky) (fabs t_1))
     (if (<= (sin kx) -0.5)
       (fabs (* (sin ky) t_1))
       (if (<= (sin kx) -0.005)
         (* t_1 (fabs (sin ky)))
         (if (<= (sin kx) 2e-19)
           (sin th)
           (* (sin th) (/ (sin ky) (sin kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / sin(kx);
	double tmp;
	if (sin(kx) <= -0.82) {
		tmp = sin(ky) * fabs(t_1);
	} else if (sin(kx) <= -0.5) {
		tmp = fabs((sin(ky) * t_1));
	} else if (sin(kx) <= -0.005) {
		tmp = t_1 * fabs(sin(ky));
	} else if (sin(kx) <= 2e-19) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / sin(kx)
    if (sin(kx) <= (-0.82d0)) then
        tmp = sin(ky) * abs(t_1)
    else if (sin(kx) <= (-0.5d0)) then
        tmp = abs((sin(ky) * t_1))
    else if (sin(kx) <= (-0.005d0)) then
        tmp = t_1 * abs(sin(ky))
    else if (sin(kx) <= 2d-19) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.sin(kx);
	double tmp;
	if (Math.sin(kx) <= -0.82) {
		tmp = Math.sin(ky) * Math.abs(t_1);
	} else if (Math.sin(kx) <= -0.5) {
		tmp = Math.abs((Math.sin(ky) * t_1));
	} else if (Math.sin(kx) <= -0.005) {
		tmp = t_1 * Math.abs(Math.sin(ky));
	} else if (Math.sin(kx) <= 2e-19) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.sin(kx)
	tmp = 0
	if math.sin(kx) <= -0.82:
		tmp = math.sin(ky) * math.fabs(t_1)
	elif math.sin(kx) <= -0.5:
		tmp = math.fabs((math.sin(ky) * t_1))
	elif math.sin(kx) <= -0.005:
		tmp = t_1 * math.fabs(math.sin(ky))
	elif math.sin(kx) <= 2e-19:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / sin(kx))
	tmp = 0.0
	if (sin(kx) <= -0.82)
		tmp = Float64(sin(ky) * abs(t_1));
	elseif (sin(kx) <= -0.5)
		tmp = abs(Float64(sin(ky) * t_1));
	elseif (sin(kx) <= -0.005)
		tmp = Float64(t_1 * abs(sin(ky)));
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / sin(kx);
	tmp = 0.0;
	if (sin(kx) <= -0.82)
		tmp = sin(ky) * abs(t_1);
	elseif (sin(kx) <= -0.5)
		tmp = abs((sin(ky) * t_1));
	elseif (sin(kx) <= -0.005)
		tmp = t_1 * abs(sin(ky));
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.82], N[(N[Sin[ky], $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -0.5], N[Abs[N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(t$95$1 * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-19], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 kx) < -0.819999999999999951

   1. Initial program 99.5%

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

      1. associate-*l/ 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l/ 99.3%

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

      4. +-commutative 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in ky around 0 2.3%

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

      1. add-sqr-sqrt 0.7%

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

      2. sqrt-unprod 40.3%

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

      3. pow2 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. unpow2 40.3%

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

      2. rem-sqrt-square 40.2%

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

   8. Simplified 40.2%

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

   if -0.819999999999999951 < (sin.f64 kx) < -0.5

   1. Initial program 99.3%

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

   2. Taylor expanded in ky around 0 12.8%

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

      1. associate-*l/ 12.8%

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

      2. *-commutative 12.8%

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

      3. add-sqr-sqrt 6.8%

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

      4. sqrt-unprod 44.9%

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

      5. pow2 44.9%

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

      6. associate-/l* 45.1%

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

      7. associate-/r/ 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. unpow2 44.9%

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

      2. rem-sqrt-square 67.5%

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

      3. *-commutative 67.5%

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

   6. Simplified 67.5%

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

   if -0.5 < (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.5%

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

      1. associate-*l/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 15.9%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-prod 24.0%

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

      3. rem-sqrt-square 38.7%

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

   6. Applied egg-rr 38.8%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 2e-19

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. unpow2 86.8%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 44.6%

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

   if 2e-19 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 52.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.82:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -0.5:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \left|\sin ky\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 5: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \leq -0.025:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin th) -0.025)
   (sin th)
   (if (<= (sin th) 1e-5)
     (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
     (* (sin ky) (fabs (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(th) <= -0.025) {
		tmp = sin(th);
	} else if (sin(th) <= 1e-5) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(ky) * fabs((sin(th) / sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(th) <= -0.025) {
		tmp = Math.sin(th);
	} else if (Math.sin(th) <= 1e-5) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = Math.sin(ky) * Math.abs((Math.sin(th) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(th) <= -0.025:
		tmp = math.sin(th)
	elif math.sin(th) <= 1e-5:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = math.sin(ky) * math.fabs((math.sin(th) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(th) <= -0.025)
		tmp = sin(th);
	elseif (sin(th) <= 1e-5)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(ky) * abs(Float64(sin(th) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(th) <= -0.025)
		tmp = sin(th);
	elseif (sin(th) <= 1e-5)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = sin(ky) * abs((sin(th) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[th], $MachinePrecision], -0.025], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[th], $MachinePrecision], 1e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[Abs[N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 th) < -0.025000000000000001

   1. Initial program 91.2%

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

      1. +-commutative 91.2%

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

      2. unpow2 91.2%

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

      3. unpow2 91.2%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 34.1%

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

   if -0.025000000000000001 < (sin.f64 th) < 1.00000000000000008e-5

   1. Initial program 96.7%

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

      1. associate-*l/ 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*l/ 96.6%

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

      4. +-commutative 96.6%

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

      5. unpow2 96.6%

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

      6. unpow2 96.6%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 94.6%

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

      1. *-commutative 94.6%

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

      2. +-commutative 94.6%

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

   6. Simplified 94.6%

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

      1. add-log-exp 18.5%

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

      2. *-commutative 18.5%

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

      3. exp-prod 19.3%

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

      4. *-commutative 19.3%

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

      5. sqrt-div 19.3%

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

      6. metadata-eval 19.3%

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

      7. +-commutative 19.3%

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

      8. unpow2 19.3%

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

      9. unpow2 19.3%

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

      10. hypot-udef 19.3%

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

      11. un-div-inv 19.3%

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

   8. Applied egg-rr 19.3%

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

      1. log-pow 73.2%

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

      2. hypot-def 72.4%

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

      3. unpow2 72.4%

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

      4. unpow2 72.4%

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

      5. +-commutative 72.4%

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

      6. unpow2 72.4%

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

      7. unpow2 72.4%

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

      8. hypot-def 73.2%

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

      9. rem-log-exp 98.5%

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

   10. Simplified 98.5%

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

       1. expm1-log1p-u 98.5%

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

       2. expm1-udef 19.3%

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

       3. div-inv 19.3%

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

       4. associate-*l* 19.3%

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

       5. associate-/r/ 19.3%

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

       6. clear-num 19.3%

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

   12. Applied egg-rr 19.3%

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

       1. expm1-def 98.6%

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

       2. expm1-log1p 98.6%

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

   14. Simplified 98.6%

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

   if 1.00000000000000008e-5 < (sin.f64 th)

   1. Initial program 90.0%

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

      1. associate-*l/ 90.0%

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

      2. *-commutative 90.0%

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

      3. associate-*l/ 90.1%

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

      4. +-commutative 90.1%

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

      5. unpow2 90.1%

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

      6. unpow2 90.1%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 16.8%

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

      1. add-sqr-sqrt 15.4%

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

      2. sqrt-unprod 41.2%

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

      3. pow2 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. unpow2 41.2%

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

      2. rem-sqrt-square 45.9%

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

   8. Simplified 45.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \leq -0.025:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]

Alternative 6: 78.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -1e-7)
     (* (/ (sin ky) t_1) th)
     (if (<= (sin ky) 4e-15) (* (sin th) (/ ky t_1)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) <= -1e-7) {
		tmp = (sin(ky) / t_1) * th;
	} else if (sin(ky) <= 4e-15) {
		tmp = sin(th) * (ky / t_1);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) <= -1e-7) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (Math.sin(ky) <= 4e-15) {
		tmp = Math.sin(th) * (ky / t_1);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) <= -1e-7:
		tmp = (math.sin(ky) / t_1) * th
	elif math.sin(ky) <= 4e-15:
		tmp = math.sin(th) * (ky / t_1)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) <= -1e-7)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (sin(ky) <= 4e-15)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) <= -1e-7)
		tmp = (sin(ky) / t_1) * th;
	elseif (sin(ky) <= 4e-15)
		tmp = sin(th) * (ky / t_1);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-7], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-15], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -9.9999999999999995e-8

   1. Initial program 99.5%

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

      1. associate-*l/ 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l/ 99.5%

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

      4. +-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 58.1%

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

      1. *-commutative 58.1%

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

      2. +-commutative 58.1%

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

   6. Simplified 58.1%

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

      1. add-log-exp 4.7%

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

      2. *-commutative 4.7%

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

      3. exp-prod 4.7%

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

      4. *-commutative 4.7%

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

      5. sqrt-div 4.7%

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

      6. metadata-eval 4.7%

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

      7. +-commutative 4.7%

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

      8. unpow2 4.7%

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

      9. unpow2 4.7%

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

      10. hypot-udef 4.7%

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

      11. un-div-inv 4.7%

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

   8. Applied egg-rr 4.7%

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

      1. log-pow 57.6%

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

      2. hypot-def 57.6%

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

      3. unpow2 57.6%

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

      4. unpow2 57.6%

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

      5. +-commutative 57.6%

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

      6. unpow2 57.6%

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

      7. unpow2 57.6%

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

      8. hypot-def 57.6%

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

      9. rem-log-exp 58.3%

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

   10. Simplified 58.3%

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

       1. expm1-log1p-u 57.8%

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

       2. expm1-udef 4.5%

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

       3. div-inv 4.5%

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

       4. associate-*l* 4.5%

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

       5. associate-/r/ 4.5%

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

       6. clear-num 4.5%

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

   12. Applied egg-rr 4.5%

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

       1. expm1-def 57.9%

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

       2. expm1-log1p 58.4%

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

   14. Simplified 58.4%

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

   if -9.9999999999999995e-8 < (sin.f64 ky) < 4.0000000000000003e-15

   1. Initial program 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 99.7%

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

   if 4.0000000000000003e-15 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 59.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7: 44.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (fabs (* (sin ky) (/ (sin th) (sin kx))))
   (if (<= (sin kx) 2e-19) (sin th) (* (sin th) (/ (sin ky) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = fabs((sin(ky) * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 2e-19) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.005d0)) then
        tmp = abs((sin(ky) * (sin(th) / sin(kx))))
    else if (sin(kx) <= 2d-19) then
        tmp = sin(th)
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = Math.abs((Math.sin(ky) * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 2e-19) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = math.fabs((math.sin(ky) * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 2e-19:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = abs(Float64(sin(ky) * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = abs((sin(ky) * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 2e-19)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[Abs[N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-19], N[Sin[th], $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. Taylor expanded in ky around 0 9.8%

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

      1. associate-*l/ 9.8%

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

      2. *-commutative 9.8%

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

      3. add-sqr-sqrt 7.6%

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

      4. sqrt-unprod 23.9%

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

      5. pow2 23.9%

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

      6. associate-/l* 23.9%

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

      7. associate-/r/ 23.9%

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

   4. Applied egg-rr 23.9%

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

      1. unpow2 23.9%

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

      2. rem-sqrt-square 34.3%

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

      3. *-commutative 34.3%

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

   6. Simplified 34.3%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 2e-19

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. unpow2 86.8%

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

      3. unpow2 86.8%

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

      4. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 44.6%

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

   if 2e-19 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0 52.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 8: 42.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-94) (* (sin th) (/ (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-94) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 4d-94) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 4e-94) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 4e-94:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 4e-94)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 4e-94)
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-94], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.9999999999999998e-94

   1. Initial program 90.6%

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

   2. Taylor expanded in ky around 0 27.2%

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

   if 3.9999999999999998e-94 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 57.4%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 33.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-211}:\\ \;\;\;\;\left|th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-211) (fabs (* th (/ ky (sin kx)))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-211) {
		tmp = fabs((th * (ky / sin(kx))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-211) then
        tmp = abs((th * (ky / sin(kx))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-211) {
		tmp = Math.abs((th * (ky / Math.sin(kx))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-211:
		tmp = math.fabs((th * (ky / math.sin(kx))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-211)
		tmp = abs(Float64(th * Float64(ky / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-211)
		tmp = abs((th * (ky / sin(kx))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-211], N[Abs[N[(th * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2.00000000000000017e-211

   1. Initial program 93.1%

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

      1. associate-*l/ 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*l/ 93.0%

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

      4. +-commutative 93.0%

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

      5. unpow2 93.0%

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

      6. unpow2 93.0%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 52.4%

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

      1. *-commutative 52.4%

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

      2. +-commutative 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in ky around 0 14.8%

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

      1. associate-/l* 15.3%

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

   9. Simplified 15.3%

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

       1. add-sqr-sqrt 13.1%

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

       2. sqrt-unprod 14.1%

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

       3. pow2 14.1%

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

       4. associate-/r/ 14.1%

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

       5. associate-*l/ 14.1%

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

   11. Applied egg-rr 14.1%

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

       1. unpow2 14.1%

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

       2. rem-sqrt-square 16.2%

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

       3. associate-*l/ 16.6%

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

       4. *-commutative 16.6%

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

   13. Simplified 16.6%

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

   if 2.00000000000000017e-211 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. unpow2 94.6%

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

      3. unpow2 94.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-211}:\\ \;\;\;\;\left|th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10: 40.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-94) (* (sin th) (/ ky (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-94) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 4d-94) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 4e-94) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 4e-94:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 4e-94)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 4e-94)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-94], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.9999999999999998e-94

   1. Initial program 90.6%

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

   2. Taylor expanded in ky around 0 24.8%

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

   if 3.9999999999999998e-94 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 57.4%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 40.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 4e-94) (/ ky (/ (sin kx) (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 4e-94) {
		tmp = ky / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 4d-94) then
        tmp = ky / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 4e-94) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 4e-94:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 4e-94)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 4e-94)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-94], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 3.9999999999999998e-94

   1. Initial program 90.6%

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

      1. +-commutative 90.6%

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

      2. unpow2 90.6%

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

      3. unpow2 90.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 24.3%

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

      1. associate-/l* 24.8%

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

   6. Simplified 24.8%

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

   if 3.9999999999999998e-94 < (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. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 57.4%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 31.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-211}:\\ \;\;\;\;\frac{th}{\frac{kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-211) (/ th (/ kx (sin ky))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-211) {
		tmp = th / (kx / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-211) then
        tmp = th / (kx / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-211) {
		tmp = th / (kx / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-211:
		tmp = th / (kx / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-211)
		tmp = Float64(th / Float64(kx / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-211)
		tmp = th / (kx / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-211], N[(th / N[(kx / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2.00000000000000017e-211

   1. Initial program 93.1%

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

      1. associate-*l/ 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*l/ 93.0%

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

      4. +-commutative 93.0%

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

      5. unpow2 93.0%

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

      6. unpow2 93.0%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in ky around 0 24.1%

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

   5. Taylor expanded in th around 0 16.2%

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

      1. associate-/l* 16.8%

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

   7. Simplified 16.8%

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

   8. Taylor expanded in kx around 0 14.1%

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

      1. associate-/l* 14.7%

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

   10. Simplified 14.7%

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

   if 2.00000000000000017e-211 < (sin.f64 ky)

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. unpow2 94.6%

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

      3. unpow2 94.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 52.4%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-211}:\\ \;\;\;\;\frac{th}{\frac{kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 27.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.2e-10) (sin th) (* ky (/ th (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.2e-10) {
		tmp = sin(th);
	} else {
		tmp = ky * (th / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 8.2d-10) then
        tmp = sin(th)
    else
        tmp = ky * (th / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.2e-10) {
		tmp = Math.sin(th);
	} else {
		tmp = ky * (th / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8.2e-10:
		tmp = math.sin(th)
	else:
		tmp = ky * (th / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.2e-10)
		tmp = sin(th);
	else
		tmp = Float64(ky * Float64(th / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8.2e-10)
		tmp = sin(th);
	else
		tmp = ky * (th / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.2e-10], N[Sin[th], $MachinePrecision], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 8.1999999999999996e-10

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. unpow2 91.6%

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

      3. unpow2 91.6%

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

      4. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 32.1%

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

   if 8.1999999999999996e-10 < kx

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l/ 99.3%

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

      4. +-commutative 99.3%

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

      5. unpow2 99.3%

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

      6. unpow2 99.3%

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

      7. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. +-commutative 57.2%

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

   6. Simplified 57.2%

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

   7. Taylor expanded in ky around 0 23.2%

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

      1. associate-/l* 23.2%

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

   9. Simplified 23.2%

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

       1. clear-num 23.2%

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

       2. associate-/r/ 23.2%

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

       3. clear-num 23.2%

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

   11. Applied egg-rr 23.2%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \end{array} \]

Alternative 14: 31.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -70000 \lor \neg \left(ky \leq 2.1 \cdot 10^{-210}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (or (<= ky -70000.0) (not (<= ky 2.1e-210))) (sin th) (* th (/ ky kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -70000.0) || !(ky <= 2.1e-210)) {
		tmp = sin(th);
	} else {
		tmp = th * (ky / kx);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((ky <= (-70000.0d0)) .or. (.not. (ky <= 2.1d-210))) then
        tmp = sin(th)
    else
        tmp = th * (ky / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((ky <= -70000.0) || !(ky <= 2.1e-210)) {
		tmp = Math.sin(th);
	} else {
		tmp = th * (ky / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (ky <= -70000.0) or not (ky <= 2.1e-210):
		tmp = math.sin(th)
	else:
		tmp = th * (ky / kx)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((ky <= -70000.0) || !(ky <= 2.1e-210))
		tmp = sin(th);
	else
		tmp = Float64(th * Float64(ky / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((ky <= -70000.0) || ~((ky <= 2.1e-210)))
		tmp = sin(th);
	else
		tmp = th * (ky / kx);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[Or[LessEqual[ky, -70000.0], N[Not[LessEqual[ky, 2.1e-210]], $MachinePrecision]], N[Sin[th], $MachinePrecision], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < -7e4 or 2.10000000000000016e-210 < ky

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 32.7%

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

   if -7e4 < ky < 2.10000000000000016e-210

   1. Initial program 86.0%

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

      1. associate-*l/ 84.2%

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

      2. *-commutative 84.2%

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

      3. associate-*l/ 85.9%

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

      4. +-commutative 85.9%

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

      5. unpow2 85.9%

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

      6. unpow2 85.9%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 47.8%

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

      1. *-commutative 47.8%

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

      2. +-commutative 47.8%

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

   6. Simplified 47.8%

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

   7. Taylor expanded in ky around 0 28.0%

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

      1. associate-/l* 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in kx around 0 24.2%

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

       1. associate-/l* 25.4%

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

   12. Simplified 25.4%

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

       1. associate-/r/ 25.5%

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

   14. Applied egg-rr 25.5%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -70000 \lor \neg \left(ky \leq 2.1 \cdot 10^{-210}\right):\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \end{array} \]

Alternative 15: 22.2% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -70000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -70000.0) th (if (<= ky 2.4e-118) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -70000.0) {
		tmp = th;
	} else if (ky <= 2.4e-118) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-70000.0d0)) then
        tmp = th
    else if (ky <= 2.4d-118) then
        tmp = th * (ky / kx)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -70000.0) {
		tmp = th;
	} else if (ky <= 2.4e-118) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -70000.0:
		tmp = th
	elif ky <= 2.4e-118:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -70000.0)
		tmp = th;
	elseif (ky <= 2.4e-118)
		tmp = Float64(th * Float64(ky / kx));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -70000.0)
		tmp = th;
	elseif (ky <= 2.4e-118)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -70000.0], th, If[LessEqual[ky, 2.4e-118], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -7e4 or 2.4000000000000001e-118 < ky

   1. Initial program 99.6%

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

      1. associate-*l/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l/ 99.6%

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

      4. +-commutative 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 54.5%

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

      1. *-commutative 54.5%

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

      2. +-commutative 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in kx around 0 18.4%

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

   if -7e4 < ky < 2.4000000000000001e-118

   1. Initial program 82.1%

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

      1. associate-*l/ 80.7%

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

      2. *-commutative 80.7%

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

      3. associate-*l/ 82.0%

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

      4. +-commutative 82.0%

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

      5. unpow2 82.0%

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

      6. unpow2 82.0%

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

      7. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 42.9%

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

      1. *-commutative 42.9%

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

      2. +-commutative 42.9%

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

   6. Simplified 42.9%

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

   7. Taylor expanded in ky around 0 26.6%

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

      1. associate-/l* 27.5%

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

   9. Simplified 27.5%

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

   10. Taylor expanded in kx around 0 22.7%

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

       1. associate-/l* 23.6%

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

   12. Simplified 23.6%

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

       1. associate-/r/ 23.7%

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

   14. Applied egg-rr 23.7%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -70000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 16: 13.4% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.7%

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

   1. associate-*l/ 93.2%

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

   2. *-commutative 93.2%

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

   3. associate-*l/ 93.7%

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

   4. +-commutative 93.7%

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

   5. unpow2 93.7%

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

   6. unpow2 93.7%

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

   7. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 50.6%

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

   1. *-commutative 50.6%

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

   2. +-commutative 50.6%

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

6. Simplified 50.6%

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

7. Taylor expanded in kx around 0 13.5%

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

8. Final simplification 13.5%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023333 
(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)))