Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 25.1s

Alternatives: 19

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: 94.0% 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 92.7%

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

   1. +-commutative 92.7%

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

   2. unpow2 92.7%

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

   3. unpow2 92.7%

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

   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: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \sin ky \cdot \frac{th}{t_1}\\ \mathbf{if}\;\sin ky \leq -0.35:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))) (t_2 (* (sin ky) (/ th t_1))))
   (if (<= (sin ky) -0.35)
     t_2
     (if (<= (sin ky) -0.02)
       (fabs (sin th))
       (if (<= (sin ky) -5e-7)
         t_2
         (if (<= (sin ky) 0.02) (* (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 t_2 = sin(ky) * (th / t_1);
	double tmp;
	if (sin(ky) <= -0.35) {
		tmp = t_2;
	} else if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-7) {
		tmp = t_2;
	} else if (sin(ky) <= 0.02) {
		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 t_2 = Math.sin(ky) * (th / t_1);
	double tmp;
	if (Math.sin(ky) <= -0.35) {
		tmp = t_2;
	} else if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-7) {
		tmp = t_2;
	} else if (Math.sin(ky) <= 0.02) {
		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))
	t_2 = math.sin(ky) * (th / t_1)
	tmp = 0
	if math.sin(ky) <= -0.35:
		tmp = t_2
	elif math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-7:
		tmp = t_2
	elif math.sin(ky) <= 0.02:
		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))
	t_2 = Float64(sin(ky) * Float64(th / t_1))
	tmp = 0.0
	if (sin(ky) <= -0.35)
		tmp = t_2;
	elseif (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-7)
		tmp = t_2;
	elseif (sin(ky) <= 0.02)
		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));
	t_2 = sin(ky) * (th / t_1);
	tmp = 0.0;
	if (sin(ky) <= -0.35)
		tmp = t_2;
	elseif (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-7)
		tmp = t_2;
	elseif (sin(ky) <= 0.02)
		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]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.35], t$95$2, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-7], t$95$2, If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -0.34999999999999998 or -0.0200000000000000004 < (sin.f64 ky) < -4.99999999999999977e-7

   1. Initial program 99.7%

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

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in th around 0 60.8%

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

      1. expm1-log1p-u 60.0%

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

      2. expm1-udef 3.7%

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

      3. div-inv 3.7%

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

      4. *-commutative 3.7%

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

      5. associate-*l* 3.7%

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

      6. div-inv 3.7%

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

   6. Applied egg-rr 3.7%

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

      1. expm1-def 60.4%

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

      2. expm1-log1p 61.2%

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

      3. *-commutative 61.2%

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

      4. associate-*l/ 60.8%

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

      5. associate-*r/ 61.2%

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

   8. Simplified 61.2%

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

   if -0.34999999999999998 < (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.3%

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

      1. clear-num 2.3%

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

      2. inv-pow 2.3%

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

   6. Applied egg-rr 2.3%

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

      1. unpow-1 2.3%

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

   8. Simplified 2.3%

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

      1. add-sqr-sqrt 0.6%

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

      2. sqrt-unprod 38.7%

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

      3. pow2 38.7%

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

      4. un-div-inv 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. unpow2 38.9%

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

       2. rem-sqrt-square 51.1%

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

       3. associate-/r/ 51.2%

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

       4. *-inverses 51.2%

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

       5. *-lft-identity 51.2%

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

   12. Simplified 51.2%

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

   if -4.99999999999999977e-7 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 84.9%

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

      1. associate-*l/ 84.1%

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

      2. +-commutative 84.1%

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

      3. unpow2 84.1%

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

      4. unpow2 84.1%

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

      5. hypot-def 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in ky around 0 91.5%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 30.9%

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

      3. associate-/l* 30.9%

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

      4. associate-/r/ 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. expm1-def 97.9%

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

      2. expm1-log1p 97.9%

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

      3. associate-*l/ 91.5%

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

      4. associate-*r/ 98.0%

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

   8. Simplified 98.0%

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

   if 0.0200000000000000004 < (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 56.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.35:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.35)
     (* (/ (sin ky) t_1) th)
     (if (<= (sin ky) -0.02)
       (fabs (sin th))
       (if (<= (sin ky) -5e-7)
         (* (sin ky) (/ th t_1))
         (if (<= (sin ky) 0.02) (* (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) <= -0.35) {
		tmp = (sin(ky) / t_1) * th;
	} else if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-7) {
		tmp = sin(ky) * (th / t_1);
	} else if (sin(ky) <= 0.02) {
		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) <= -0.35) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-7) {
		tmp = Math.sin(ky) * (th / t_1);
	} else if (Math.sin(ky) <= 0.02) {
		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) <= -0.35:
		tmp = (math.sin(ky) / t_1) * th
	elif math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-7:
		tmp = math.sin(ky) * (th / t_1)
	elif math.sin(ky) <= 0.02:
		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) <= -0.35)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-7)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	elseif (sin(ky) <= 0.02)
		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) <= -0.35)
		tmp = (sin(ky) / t_1) * th;
	elseif (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-7)
		tmp = sin(ky) * (th / t_1);
	elseif (sin(ky) <= 0.02)
		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], -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-7], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.34999999999999998

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 59.7%

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

      1. associate-/l* 59.7%

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

      2. associate-/r/ 59.7%

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

   6. Applied egg-rr 59.7%

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

   if -0.34999999999999998 < (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.8%

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

      1. associate-*l/ 99.6%

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

      2. associate-*r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. unpow2 99.7%

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

      6. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in kx around 0 2.3%

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

      1. clear-num 2.3%

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

      2. inv-pow 2.3%

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

   6. Applied egg-rr 2.3%

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

      1. unpow-1 2.3%

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

   8. Simplified 2.3%

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

      1. add-sqr-sqrt 0.6%

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

      2. sqrt-unprod 38.7%

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

      3. pow2 38.7%

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

      4. un-div-inv 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. unpow2 38.9%

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

       2. rem-sqrt-square 51.1%

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

       3. associate-/r/ 51.2%

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

       4. *-inverses 51.2%

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

       5. *-lft-identity 51.2%

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

   12. Simplified 51.2%

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

   if -0.0200000000000000004 < (sin.f64 ky) < -4.99999999999999977e-7

   1. Initial program 100.0%

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

      1. associate-*l/ 89.2%

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

      2. +-commutative 89.2%

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

      3. unpow2 89.2%

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

      4. unpow2 89.2%

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

      5. hypot-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in th around 0 89.2%

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

      1. expm1-log1p-u 89.2%

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

      2. expm1-udef 9.5%

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

      3. div-inv 9.5%

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

      4. *-commutative 9.5%

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

      5. associate-*l* 9.5%

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

      6. div-inv 9.5%

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

   6. Applied egg-rr 9.5%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l/ 89.2%

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

      5. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

   if -4.99999999999999977e-7 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 84.9%

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

      1. associate-*l/ 84.1%

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

      2. +-commutative 84.1%

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

      3. unpow2 84.1%

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

      4. unpow2 84.1%

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

      5. hypot-def 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in ky around 0 91.5%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 30.9%

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

      3. associate-/l* 30.9%

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

      4. associate-/r/ 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. expm1-def 97.9%

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

      2. expm1-log1p 97.9%

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

      3. associate-*l/ 91.5%

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

      4. associate-*r/ 98.0%

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

   8. Simplified 98.0%

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

   if 0.0200000000000000004 < (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 56.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.34:\\ \;\;\;\;\frac{\frac{\sin ky}{t_1}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (<= (sin ky) -0.34)
     (/ (/ (sin ky) t_1) (+ (/ 1.0 th) (* th 0.16666666666666666)))
     (if (<= (sin ky) -0.02)
       (fabs (sin th))
       (if (<= (sin ky) -5e-7)
         (* (sin ky) (/ th t_1))
         (if (<= (sin ky) 0.02) (* (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) <= -0.34) {
		tmp = (sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	} else if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -5e-7) {
		tmp = sin(ky) * (th / t_1);
	} else if (sin(ky) <= 0.02) {
		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) <= -0.34) {
		tmp = (Math.sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	} else if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -5e-7) {
		tmp = Math.sin(ky) * (th / t_1);
	} else if (Math.sin(ky) <= 0.02) {
		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) <= -0.34:
		tmp = (math.sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666))
	elif math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -5e-7:
		tmp = math.sin(ky) * (th / t_1)
	elif math.sin(ky) <= 0.02:
		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) <= -0.34)
		tmp = Float64(Float64(sin(ky) / t_1) / Float64(Float64(1.0 / th) + Float64(th * 0.16666666666666666)));
	elseif (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-7)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	elseif (sin(ky) <= 0.02)
		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) <= -0.34)
		tmp = (sin(ky) / t_1) / ((1.0 / th) + (th * 0.16666666666666666));
	elseif (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -5e-7)
		tmp = sin(ky) * (th / t_1);
	elseif (sin(ky) <= 0.02)
		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], -0.34], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(N[(1.0 / th), $MachinePrecision] + N[(th * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-7], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (sin.f64 ky) < -0.340000000000000024

   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. Step-by-step derivation

      1. associate-/r/ 99.6%

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

      2. div-inv 99.5%

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

      3. associate-/r* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 58.7%

      \[\leadsto \frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\color{blue}{\frac{1}{th} + 0.16666666666666666 \cdot th}} \]
   7. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto \frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{th} + \color{blue}{th \cdot 0.16666666666666666}} \]

   8. Simplified 58.7%

      \[\leadsto \frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\color{blue}{\frac{1}{th} + th \cdot 0.16666666666666666}} \]

   if -0.340000000000000024 < (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.8%

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

      1. associate-*l/ 99.7%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. unpow2 99.8%

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

      5. unpow2 99.8%

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

      6. hypot-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in kx around 0 2.3%

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

      1. clear-num 2.3%

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

      2. inv-pow 2.3%

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

   6. Applied egg-rr 2.3%

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

      1. unpow-1 2.3%

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

   8. Simplified 2.3%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 41.3%

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

      3. pow2 41.3%

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

      4. un-div-inv 41.6%

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

   10. Applied egg-rr 41.6%

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

       1. unpow2 41.6%

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

       2. rem-sqrt-square 54.6%

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

       3. associate-/r/ 54.7%

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

       4. *-inverses 54.7%

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

       5. *-lft-identity 54.7%

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

   12. Simplified 54.7%

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

   if -0.0200000000000000004 < (sin.f64 ky) < -4.99999999999999977e-7

   1. Initial program 100.0%

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

      1. associate-*l/ 89.2%

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

      2. +-commutative 89.2%

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

      3. unpow2 89.2%

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

      4. unpow2 89.2%

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

      5. hypot-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in th around 0 89.2%

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

      1. expm1-log1p-u 89.2%

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

      2. expm1-udef 9.5%

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

      3. div-inv 9.5%

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

      4. *-commutative 9.5%

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

      5. associate-*l* 9.5%

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

      6. div-inv 9.5%

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

   6. Applied egg-rr 9.5%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l/ 89.2%

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

      5. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

   if -4.99999999999999977e-7 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 84.9%

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

      1. associate-*l/ 84.1%

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

      2. +-commutative 84.1%

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

      3. unpow2 84.1%

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

      4. unpow2 84.1%

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

      5. hypot-def 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in ky around 0 91.5%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 30.9%

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

      3. associate-/l* 30.9%

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

      4. associate-/r/ 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. expm1-def 97.9%

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

      2. expm1-log1p 97.9%

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

      3. associate-*l/ 91.5%

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

      4. associate-*r/ 98.0%

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

   8. Simplified 98.0%

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

   if 0.0200000000000000004 < (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 56.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.34:\\ \;\;\;\;\frac{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-277}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-39)
   (fabs (sin th))
   (if (<= (sin ky) -1e-277)
     (/ (sin th) (+ (/ (sin kx) ky) (* ky (/ 0.5 kx))))
     (if (<= (sin ky) 5e-70)
       (* (sin th) (fabs (/ (sin ky) (sin kx))))
       (/ (* (sin ky) (sin th)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-39) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= -1e-277) {
		tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5 / kx)));
	} else if (sin(ky) <= 5e-70) {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	} else {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-2d-39)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= (-1d-277)) then
        tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5d0 / kx)))
    else if (sin(ky) <= 5d-70) then
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    else
        tmp = (sin(ky) * sin(th)) / sin(ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -2e-39) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= -1e-277) {
		tmp = Math.sin(th) / ((Math.sin(kx) / ky) + (ky * (0.5 / kx)));
	} else if (Math.sin(ky) <= 5e-70) {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	} else {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-39:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= -1e-277:
		tmp = math.sin(th) / ((math.sin(kx) / ky) + (ky * (0.5 / kx)))
	elif math.sin(ky) <= 5e-70:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	else:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -1e-277)
		tmp = Float64(sin(th) / Float64(Float64(sin(kx) / ky) + Float64(ky * Float64(0.5 / kx))));
	elseif (sin(ky) <= 5e-70)
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= -1e-277)
		tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5 / kx)));
	elseif (sin(ky) <= 5e-70)
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	else
		tmp = (sin(ky) * sin(th)) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-39], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], -1e-277], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision] + N[(ky * N[(0.5 / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-70], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1.99999999999999986e-39

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.9%

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

      1. clear-num 2.9%

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

      2. inv-pow 2.9%

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

   6. Applied egg-rr 2.9%

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

      1. unpow-1 2.9%

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

   8. Simplified 2.9%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 25.0%

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

      3. pow2 25.0%

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

      4. un-div-inv 25.0%

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

   10. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. rem-sqrt-square 38.9%

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

       3. associate-/r/ 39.0%

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

       4. *-inverses 39.0%

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

       5. *-lft-identity 39.0%

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

   12. Simplified 39.0%

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

   if -1.99999999999999986e-39 < (sin.f64 ky) < -9.99999999999999969e-278

   1. Initial program 90.1%

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

      1. +-commutative 90.1%

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

      2. unpow2 90.1%

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

      3. unpow2 90.1%

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

      4. hypot-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 95.3%

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

      2. associate-*r/ 99.5%

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

      3. expm1-log1p-u 99.5%

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

      4. expm1-udef 22.4%

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

   5. Applied egg-rr 22.4%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r/ 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in ky around 0 51.3%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky} + \left(0.5 \cdot \frac{1}{\sin kx} - -0.16666666666666666 \cdot \sin kx\right) \cdot ky}} \]
   9. Step-by-step derivation

      1. *-commutative 51.3%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + \color{blue}{ky \cdot \left(0.5 \cdot \frac{1}{\sin kx} - -0.16666666666666666 \cdot \sin kx\right)}} \]

      2. cancel-sign-sub-inv 51.3%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\sin kx} + \left(--0.16666666666666666\right) \cdot \sin kx\right)}} \]

      3. associate-*r/ 51.3%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{\sin kx}} + \left(--0.16666666666666666\right) \cdot \sin kx\right)} \]

      4. metadata-eval 51.3%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{\color{blue}{0.5}}{\sin kx} + \left(--0.16666666666666666\right) \cdot \sin kx\right)} \]

      5. metadata-eval 51.3%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{0.5}{\sin kx} + \color{blue}{0.16666666666666666} \cdot \sin kx\right)} \]

   10. Simplified 51.3%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{0.5}{\sin kx} + 0.16666666666666666 \cdot \sin kx\right)}} \]

   11. Taylor expanded in kx around 0 51.3%

       \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \color{blue}{\frac{0.5}{kx}}} \]

   if -9.99999999999999969e-278 < (sin.f64 ky) < 4.9999999999999998e-70

   1. Initial program 72.1%

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

      1. +-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. unpow2 72.1%

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

      4. hypot-def 99.7%

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

   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 55.5%

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

      1. add-sqr-sqrt 39.2%

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

      2. sqrt-unprod 64.7%

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

      3. pow2 64.7%

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

   6. Applied egg-rr 64.7%

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

      1. unpow2 64.7%

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

      2. rem-sqrt-square 80.4%

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

   8. Simplified 80.4%

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

   if 4.9999999999999998e-70 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 55.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-277}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \]

Alternative 6: 47.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-39)
   (fabs (sin th))
   (if (<= (sin ky) 5e-70)
     (/ (sin th) (+ (/ (sin kx) ky) (* ky (/ 0.5 kx))))
     (/ (* (sin ky) (sin th)) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-39) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-70) {
		tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5 / kx)));
	} else {
		tmp = (sin(ky) * sin(th)) / sin(ky);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-2d-39)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-70) then
        tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5d0 / kx)))
    else
        tmp = (sin(ky) * sin(th)) / sin(ky)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -2e-39) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-70) {
		tmp = Math.sin(th) / ((Math.sin(kx) / ky) + (ky * (0.5 / kx)));
	} else {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sin(ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-39:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-70:
		tmp = math.sin(th) / ((math.sin(kx) / ky) + (ky * (0.5 / kx)))
	else:
		tmp = (math.sin(ky) * math.sin(th)) / math.sin(ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-70)
		tmp = Float64(sin(th) / Float64(Float64(sin(kx) / ky) + Float64(ky * Float64(0.5 / kx))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-70)
		tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5 / kx)));
	else
		tmp = (sin(ky) * sin(th)) / sin(ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-39], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-70], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision] + N[(ky * N[(0.5 / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1.99999999999999986e-39

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.9%

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

      1. clear-num 2.9%

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

      2. inv-pow 2.9%

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

   6. Applied egg-rr 2.9%

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

      1. unpow-1 2.9%

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

   8. Simplified 2.9%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 25.0%

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

      3. pow2 25.0%

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

      4. un-div-inv 25.0%

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

   10. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. rem-sqrt-square 38.9%

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

       3. associate-/r/ 39.0%

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

       4. *-inverses 39.0%

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

       5. *-lft-identity 39.0%

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

   12. Simplified 39.0%

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

   if -1.99999999999999986e-39 < (sin.f64 ky) < 4.9999999999999998e-70

   1. Initial program 81.5%

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

      1. +-commutative 81.5%

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

      2. unpow2 81.5%

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

      3. unpow2 81.5%

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

      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. Step-by-step derivation

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 99.6%

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

      3. expm1-log1p-u 99.6%

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

      4. expm1-udef 27.8%

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

   5. Applied egg-rr 27.8%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r/ 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in ky around 0 53.4%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky} + \left(0.5 \cdot \frac{1}{\sin kx} - -0.16666666666666666 \cdot \sin kx\right) \cdot ky}} \]
   9. Step-by-step derivation

      1. *-commutative 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + \color{blue}{ky \cdot \left(0.5 \cdot \frac{1}{\sin kx} - -0.16666666666666666 \cdot \sin kx\right)}} \]

      2. cancel-sign-sub-inv 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\sin kx} + \left(--0.16666666666666666\right) \cdot \sin kx\right)}} \]

      3. associate-*r/ 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{\sin kx}} + \left(--0.16666666666666666\right) \cdot \sin kx\right)} \]

      4. metadata-eval 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{\color{blue}{0.5}}{\sin kx} + \left(--0.16666666666666666\right) \cdot \sin kx\right)} \]

      5. metadata-eval 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{0.5}{\sin kx} + \color{blue}{0.16666666666666666} \cdot \sin kx\right)} \]

   10. Simplified 53.4%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{0.5}{\sin kx} + 0.16666666666666666 \cdot \sin kx\right)}} \]

   11. Taylor expanded in kx around 0 53.4%

       \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \color{blue}{\frac{0.5}{kx}}} \]

   if 4.9999999999999998e-70 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. unpow2 99.5%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in kx around 0 55.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \]

Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. associate-*r/ 92.7%

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

   3. +-commutative 92.7%

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

   4. unpow2 92.7%

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

   5. unpow2 92.7%

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

   6. hypot-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 8: 47.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-39)
   (fabs (sin th))
   (if (<= (sin ky) 5e-70)
     (/ (sin th) (+ (/ (sin kx) ky) (* ky (/ 0.5 kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-39) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-70) {
		tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5 / 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-39)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-70) then
        tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5d0 / 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-39) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-70) {
		tmp = Math.sin(th) / ((Math.sin(kx) / ky) + (ky * (0.5 / kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -2e-39:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-70:
		tmp = math.sin(th) / ((math.sin(kx) / ky) + (ky * (0.5 / kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-70)
		tmp = Float64(sin(th) / Float64(Float64(sin(kx) / ky) + Float64(ky * Float64(0.5 / kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-70)
		tmp = sin(th) / ((sin(kx) / ky) + (ky * (0.5 / kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -2e-39], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-70], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision] + N[(ky * N[(0.5 / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1.99999999999999986e-39

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.9%

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

      1. clear-num 2.9%

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

      2. inv-pow 2.9%

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

   6. Applied egg-rr 2.9%

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

      1. unpow-1 2.9%

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

   8. Simplified 2.9%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 25.0%

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

      3. pow2 25.0%

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

      4. un-div-inv 25.0%

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

   10. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. rem-sqrt-square 38.9%

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

       3. associate-/r/ 39.0%

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

       4. *-inverses 39.0%

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

       5. *-lft-identity 39.0%

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

   12. Simplified 39.0%

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

   if -1.99999999999999986e-39 < (sin.f64 ky) < 4.9999999999999998e-70

   1. Initial program 81.5%

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

      1. +-commutative 81.5%

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

      2. unpow2 81.5%

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

      3. unpow2 81.5%

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

      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. Step-by-step derivation

      1. associate-*l/ 91.7%

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

      2. associate-*r/ 99.6%

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

      3. expm1-log1p-u 99.6%

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

      4. expm1-udef 27.8%

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

   5. Applied egg-rr 27.8%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r/ 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in ky around 0 53.4%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky} + \left(0.5 \cdot \frac{1}{\sin kx} - -0.16666666666666666 \cdot \sin kx\right) \cdot ky}} \]
   9. Step-by-step derivation

      1. *-commutative 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + \color{blue}{ky \cdot \left(0.5 \cdot \frac{1}{\sin kx} - -0.16666666666666666 \cdot \sin kx\right)}} \]

      2. cancel-sign-sub-inv 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\sin kx} + \left(--0.16666666666666666\right) \cdot \sin kx\right)}} \]

      3. associate-*r/ 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{\sin kx}} + \left(--0.16666666666666666\right) \cdot \sin kx\right)} \]

      4. metadata-eval 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{\color{blue}{0.5}}{\sin kx} + \left(--0.16666666666666666\right) \cdot \sin kx\right)} \]

      5. metadata-eval 53.4%

         \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{0.5}{\sin kx} + \color{blue}{0.16666666666666666} \cdot \sin kx\right)} \]

   10. Simplified 53.4%

       \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky} + ky \cdot \left(\frac{0.5}{\sin kx} + 0.16666666666666666 \cdot \sin kx\right)}} \]

   11. Taylor expanded in kx around 0 53.4%

       \[\leadsto \frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \color{blue}{\frac{0.5}{kx}}} \]

   if 4.9999999999999998e-70 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 54.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9: 60.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq -2.65:\\ \;\;\;\;\frac{\sin th}{1 + 0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}\\ \mathbf{elif}\;th \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 8.4 \cdot 10^{+102}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th -2.65)
   (/ (sin th) (+ 1.0 (* 0.5 (* (/ kx ky) (/ kx ky)))))
   (if (<= th 7.2e-6)
     (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
     (if (<= th 8.4e+102)
       (fabs (* (sin th) (/ (sin ky) (sin kx))))
       (/ (sin ky) (/ (sin kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= -2.65) {
		tmp = sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
	} else if (th <= 7.2e-6) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else if (th <= 8.4e+102) {
		tmp = fabs((sin(th) * (sin(ky) / sin(kx))));
	} else {
		tmp = sin(ky) / (sin(kx) / sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= -2.65) {
		tmp = Math.sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
	} else if (th <= 7.2e-6) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else if (th <= 8.4e+102) {
		tmp = Math.abs((Math.sin(th) * (Math.sin(ky) / Math.sin(kx))));
	} else {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= -2.65:
		tmp = math.sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))))
	elif th <= 7.2e-6:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	elif th <= 8.4e+102:
		tmp = math.fabs((math.sin(th) * (math.sin(ky) / math.sin(kx))))
	else:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= -2.65)
		tmp = Float64(sin(th) / Float64(1.0 + Float64(0.5 * Float64(Float64(kx / ky) * Float64(kx / ky)))));
	elseif (th <= 7.2e-6)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	elseif (th <= 8.4e+102)
		tmp = abs(Float64(sin(th) * Float64(sin(ky) / sin(kx))));
	else
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= -2.65)
		tmp = sin(th) / (1.0 + (0.5 * ((kx / ky) * (kx / ky))));
	elseif (th <= 7.2e-6)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	elseif (th <= 8.4e+102)
		tmp = abs((sin(th) * (sin(ky) / sin(kx))));
	else
		tmp = sin(ky) / (sin(kx) / sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, -2.65], N[(N[Sin[th], $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(kx / ky), $MachinePrecision] * N[(kx / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 7.2e-6], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 8.4e+102], N[Abs[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if th < -2.64999999999999991

   1. Initial program 93.3%

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

      1. +-commutative 93.3%

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

      2. unpow2 93.3%

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

      3. unpow2 93.3%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*l/ 99.5%

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

      2. associate-*r/ 99.5%

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

      3. expm1-log1p-u 99.5%

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

      4. expm1-udef 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r/ 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in kx around 0 26.5%

      \[\leadsto \frac{\sin th}{\color{blue}{1 + 0.5 \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 26.5%

         \[\leadsto \frac{\sin th}{1 + \color{blue}{\frac{0.5 \cdot {kx}^{2}}{{\sin ky}^{2}}}} \]

      2. unpow2 26.5%

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

   10. Simplified 26.5%

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

   11. Taylor expanded in ky around 0 27.1%

       \[\leadsto \frac{\sin th}{1 + \color{blue}{0.5 \cdot \frac{{kx}^{2}}{{ky}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 27.1%

          \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \frac{\color{blue}{kx \cdot kx}}{{ky}^{2}}} \]

       2. unpow2 27.1%

          \[\leadsto \frac{\sin th}{1 + 0.5 \cdot \frac{kx \cdot kx}{\color{blue}{ky \cdot ky}}} \]

       3. times-frac 29.9%

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

   13. Simplified 29.9%

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

   if -2.64999999999999991 < th < 7.19999999999999967e-6

   1. Initial program 91.2%

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

      1. associate-*l/ 90.1%

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

      2. +-commutative 90.1%

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

      3. unpow2 90.1%

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

      4. unpow2 90.1%

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

      5. hypot-def 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in th around 0 92.4%

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

      1. expm1-log1p-u 92.4%

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

      2. expm1-udef 19.7%

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

      3. div-inv 19.7%

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

      4. *-commutative 19.7%

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

      5. associate-*l* 19.7%

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

      6. div-inv 19.7%

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

   6. Applied egg-rr 19.7%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l/ 92.4%

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

      5. associate-*r/ 99.5%

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

   8. Simplified 99.5%

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

   if 7.19999999999999967e-6 < th < 8.40000000000000006e102

   1. Initial program 95.9%

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

      1. +-commutative 95.9%

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

      2. unpow2 95.9%

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

      3. unpow2 95.9%

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

      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 ky around 0 19.3%

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

      1. add-sqr-sqrt 13.3%

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

      2. sqrt-unprod 24.6%

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

      3. pow2 24.6%

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

      4. *-commutative 24.6%

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

   6. Applied egg-rr 24.6%

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

      1. unpow2 24.6%

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

      2. rem-sqrt-square 39.1%

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

   8. Simplified 39.1%

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

   if 8.40000000000000006e102 < th

   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-/r/ 93.8%

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

      2. +-commutative 93.8%

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

      3. unpow2 93.8%

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

      4. unpow2 93.8%

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

      5. hypot-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in ky around 0 33.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq -2.65:\\ \;\;\;\;\frac{\sin th}{1 + 0.5 \cdot \left(\frac{kx}{ky} \cdot \frac{kx}{ky}\right)}\\ \mathbf{elif}\;th \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 8.4 \cdot 10^{+102}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 10: 47.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -2e-39)
   (fabs (sin th))
   (if (<= (sin ky) 5e-70) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -2e-39) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-70) {
		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) <= (-2d-39)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-70) 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) <= -2e-39) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-70) {
		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) <= -2e-39:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-70:
		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) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-70)
		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) <= -2e-39)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-70)
		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], -2e-39], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-70], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1.99999999999999986e-39

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. unpow2 99.6%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.9%

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

      1. clear-num 2.9%

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

      2. inv-pow 2.9%

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

   6. Applied egg-rr 2.9%

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

      1. unpow-1 2.9%

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

   8. Simplified 2.9%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 25.0%

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

      3. pow2 25.0%

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

      4. un-div-inv 25.0%

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

   10. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. rem-sqrt-square 38.9%

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

       3. associate-/r/ 39.0%

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

       4. *-inverses 39.0%

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

       5. *-lft-identity 39.0%

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

   12. Simplified 39.0%

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

   if -1.99999999999999986e-39 < (sin.f64 ky) < 4.9999999999999998e-70

   1. Initial program 81.5%

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

      1. +-commutative 81.5%

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

      2. unpow2 81.5%

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

      3. unpow2 81.5%

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

      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 53.1%

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

   if 4.9999999999999998e-70 < (sin.f64 ky)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. unpow2 99.6%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 54.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11: 41.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-186}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -5e-186)
   (fabs (sin th))
   (if (<= (sin ky) 2e-197) (/ (sin th) (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -5e-186) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 2e-197) {
		tmp = sin(th) / (kx / 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) <= (-5d-186)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 2d-197) then
        tmp = sin(th) / (kx / 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) <= -5e-186) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 2e-197) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -5e-186:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 2e-197:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -5e-186)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-197)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -5e-186)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 2e-197)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -5e-186], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-197], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -5e-186

   1. Initial program 98.8%

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

      1. associate-*l/ 97.7%

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

      2. associate-*r/ 98.7%

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

      3. +-commutative 98.7%

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

      4. unpow2 98.7%

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

      5. unpow2 98.7%

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

      6. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 2.8%

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

      1. clear-num 2.8%

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

      2. inv-pow 2.8%

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

   6. Applied egg-rr 2.8%

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

      1. unpow-1 2.8%

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

   8. Simplified 2.8%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 22.3%

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

      3. pow2 22.3%

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

      4. un-div-inv 22.3%

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

   10. Applied egg-rr 22.3%

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

       1. unpow2 22.3%

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

       2. rem-sqrt-square 32.0%

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

       3. associate-/r/ 32.1%

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

       4. *-inverses 32.1%

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

       5. *-lft-identity 32.1%

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

   12. Simplified 32.1%

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

   if -5e-186 < (sin.f64 ky) < 2e-197

   1. Initial program 71.2%

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

      1. +-commutative 71.2%

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

      2. unpow2 71.2%

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

      3. unpow2 71.2%

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

      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 ky around 0 68.2%

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

   5. Taylor expanded in kx around 0 42.8%

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

      1. associate-/l* 44.4%

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

   7. Simplified 44.4%

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

   8. Taylor expanded in ky around 0 42.8%

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

      1. associate-/l* 44.4%

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

   10. Simplified 44.4%

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

   if 2e-197 < (sin.f64 ky)

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

      3. unpow2 97.0%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 51.4%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-186}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12: 34.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky} + 0.16666666666666666 \cdot \left(ky \cdot kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-197)
   (/ (sin th) (+ (/ kx ky) (* 0.16666666666666666 (* ky kx))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-197) {
		tmp = sin(th) / ((kx / ky) + (0.16666666666666666 * (ky * kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 2d-197) then
        tmp = sin(th) / ((kx / ky) + (0.16666666666666666d0 * (ky * kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-197) {
		tmp = Math.sin(th) / ((kx / ky) + (0.16666666666666666 * (ky * kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-197:
		tmp = math.sin(th) / ((kx / ky) + (0.16666666666666666 * (ky * kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-197)
		tmp = Float64(sin(th) / Float64(Float64(kx / ky) + Float64(0.16666666666666666 * Float64(ky * kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-197)
		tmp = sin(th) / ((kx / ky) + (0.16666666666666666 * (ky * kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-197], N[(N[Sin[th], $MachinePrecision] / N[(N[(kx / ky), $MachinePrecision] + N[(0.16666666666666666 * N[(ky * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2e-197

   1. Initial program 90.0%

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

      1. +-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. unpow2 90.0%

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

      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 32.4%

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

   5. Taylor expanded in kx around 0 18.6%

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

      1. associate-/l* 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in ky around 0 19.5%

      \[\leadsto \frac{\sin th}{\color{blue}{0.16666666666666666 \cdot \left(ky \cdot kx\right) + \frac{kx}{ky}}} \]

   if 2e-197 < (sin.f64 ky)

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

      3. unpow2 97.0%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 51.4%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky} + 0.16666666666666666 \cdot \left(ky \cdot kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13: 31.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-197) (/ (sin ky) (/ kx th)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-197) {
		tmp = sin(ky) / (kx / 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) <= 2d-197) then
        tmp = sin(ky) / (kx / 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) <= 2e-197) {
		tmp = Math.sin(ky) / (kx / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-197:
		tmp = math.sin(ky) / (kx / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-197)
		tmp = Float64(sin(ky) / Float64(kx / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-197)
		tmp = sin(ky) / (kx / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-197], N[(N[Sin[ky], $MachinePrecision] / N[(kx / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2e-197

   1. Initial program 90.0%

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

      1. +-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. unpow2 90.0%

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

      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 32.4%

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

   5. Taylor expanded in kx around 0 18.6%

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

      1. associate-/l* 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in th around 0 13.1%

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

      1. associate-/l* 13.6%

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

   10. Simplified 13.6%

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

   if 2e-197 < (sin.f64 ky)

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

      3. unpow2 97.0%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 51.4%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin ky}{\frac{kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14: 34.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-197) (/ (sin th) (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-197) {
		tmp = sin(th) / (kx / 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-197) then
        tmp = sin(th) / (kx / 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-197) {
		tmp = Math.sin(th) / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-197:
		tmp = math.sin(th) / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-197)
		tmp = Float64(sin(th) / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-197)
		tmp = sin(th) / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-197], N[(N[Sin[th], $MachinePrecision] / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2e-197

   1. Initial program 90.0%

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

      1. +-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. unpow2 90.0%

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

      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 32.4%

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

   5. Taylor expanded in kx around 0 18.6%

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

      1. associate-/l* 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in ky around 0 18.5%

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

      1. associate-/l* 19.0%

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

   10. Simplified 19.0%

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

   if 2e-197 < (sin.f64 ky)

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. unpow2 97.0%

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

      3. unpow2 97.0%

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

      4. hypot-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in kx around 0 51.4%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15: 32.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -3.7 \cdot 10^{-36}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -3.7e-36)
   (sin th)
   (if (<= ky 3.6e-193) (/ ky (/ (sin kx) th)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -3.7e-36) {
		tmp = sin(th);
	} else if (ky <= 3.6e-193) {
		tmp = ky / (sin(kx) / th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= (-3.7d-36)) then
        tmp = sin(th)
    else if (ky <= 3.6d-193) then
        tmp = ky / (sin(kx) / th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -3.7e-36) {
		tmp = Math.sin(th);
	} else if (ky <= 3.6e-193) {
		tmp = ky / (Math.sin(kx) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -3.7e-36:
		tmp = math.sin(th)
	elif ky <= 3.6e-193:
		tmp = ky / (math.sin(kx) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -3.7e-36)
		tmp = sin(th);
	elseif (ky <= 3.6e-193)
		tmp = Float64(ky / Float64(sin(kx) / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -3.7e-36)
		tmp = sin(th);
	elseif (ky <= 3.6e-193)
		tmp = ky / (sin(kx) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -3.7e-36], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.6e-193], N[(ky / N[(N[Sin[kx], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -3.70000000000000002e-36 or 3.5999999999999999e-193 < ky

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. unpow2 98.2%

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

      3. unpow2 98.2%

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

      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 30.9%

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

   if -3.70000000000000002e-36 < ky < 3.5999999999999999e-193

   1. Initial program 81.6%

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

      1. associate-*l/ 80.4%

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

      2. +-commutative 80.4%

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

      3. unpow2 80.4%

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

      4. unpow2 80.4%

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

      5. hypot-def 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in th around 0 31.6%

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

   5. Taylor expanded in ky around 0 21.3%

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

      1. associate-/l* 22.2%

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

   7. Simplified 22.2%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -3.7 \cdot 10^{-36}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16: 31.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -6.8e-7) (sin th) (if (<= ky 3.2e-185) (/ th (/ kx ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -6.8e-7) {
		tmp = sin(th);
	} else if (ky <= 3.2e-185) {
		tmp = th / (kx / 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 (ky <= (-6.8d-7)) then
        tmp = sin(th)
    else if (ky <= 3.2d-185) then
        tmp = th / (kx / 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 (ky <= -6.8e-7) {
		tmp = Math.sin(th);
	} else if (ky <= 3.2e-185) {
		tmp = th / (kx / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -6.8e-7:
		tmp = math.sin(th)
	elif ky <= 3.2e-185:
		tmp = th / (kx / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -6.8e-7)
		tmp = sin(th);
	elseif (ky <= 3.2e-185)
		tmp = Float64(th / Float64(kx / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -6.8e-7)
		tmp = sin(th);
	elseif (ky <= 3.2e-185)
		tmp = th / (kx / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -6.8e-7], N[Sin[th], $MachinePrecision], If[LessEqual[ky, 3.2e-185], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < -6.79999999999999948e-7 or 3.1999999999999997e-185 < ky

   1. Initial program 98.1%

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

      1. +-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. unpow2 98.1%

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

      4. hypot-def 99.7%

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

   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 31.7%

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

   if -6.79999999999999948e-7 < ky < 3.1999999999999997e-185

   1. Initial program 82.6%

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

      1. associate-*l/ 81.4%

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

      2. +-commutative 81.4%

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

      3. unpow2 81.4%

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

      4. unpow2 81.4%

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

      5. hypot-def 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in th around 0 32.2%

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

   5. Taylor expanded in ky around 0 20.2%

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

      1. associate-/l* 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in kx around 0 20.0%

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

      1. associate-/l* 21.1%

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

   10. Simplified 21.1%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17: 21.8% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-19}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-122}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -8e-19) th (if (<= ky 4.6e-122) (* th (/ ky kx)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -8e-19) {
		tmp = th;
	} else if (ky <= 4.6e-122) {
		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 <= (-8d-19)) then
        tmp = th
    else if (ky <= 4.6d-122) 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 <= -8e-19) {
		tmp = th;
	} else if (ky <= 4.6e-122) {
		tmp = th * (ky / kx);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -8e-19:
		tmp = th
	elif ky <= 4.6e-122:
		tmp = th * (ky / kx)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -8e-19)
		tmp = th;
	elseif (ky <= 4.6e-122)
		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 <= -8e-19)
		tmp = th;
	elseif (ky <= 4.6e-122)
		tmp = th * (ky / kx);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -8e-19], th, If[LessEqual[ky, 4.6e-122], N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -7.9999999999999998e-19 or 4.60000000000000014e-122 < 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. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 49.3%

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

   5. Taylor expanded in kx around 0 14.3%

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

   if -7.9999999999999998e-19 < ky < 4.60000000000000014e-122

   1. Initial program 80.7%

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

      1. associate-*l/ 79.6%

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

      2. +-commutative 79.6%

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

      3. unpow2 79.6%

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

      4. unpow2 79.6%

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

      5. hypot-def 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in th around 0 32.7%

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

   5. Taylor expanded in ky around 0 20.5%

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

      1. associate-/l* 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in kx around 0 21.1%

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

      1. associate-/r/ 21.3%

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

   10. Applied egg-rr 21.3%

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

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -8 \cdot 10^{-19}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-122}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 18: 21.8% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq -4.5 \cdot 10^{-19}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.12 \cdot 10^{-122}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky -4.5e-19) th (if (<= ky 1.12e-122) (/ th (/ kx ky)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -4.5e-19) {
		tmp = th;
	} else if (ky <= 1.12e-122) {
		tmp = th / (kx / ky);
	} 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 <= (-4.5d-19)) then
        tmp = th
    else if (ky <= 1.12d-122) then
        tmp = th / (kx / ky)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= -4.5e-19) {
		tmp = th;
	} else if (ky <= 1.12e-122) {
		tmp = th / (kx / ky);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= -4.5e-19:
		tmp = th
	elif ky <= 1.12e-122:
		tmp = th / (kx / ky)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= -4.5e-19)
		tmp = th;
	elseif (ky <= 1.12e-122)
		tmp = Float64(th / Float64(kx / ky));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= -4.5e-19)
		tmp = th;
	elseif (ky <= 1.12e-122)
		tmp = th / (kx / ky);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, -4.5e-19], th, If[LessEqual[ky, 1.12e-122], N[(th / N[(kx / ky), $MachinePrecision]), $MachinePrecision], th]]

Derivation

1. Split input into 2 regimes

2. if ky < -4.50000000000000013e-19 or 1.12e-122 < 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. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. hypot-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 49.3%

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

   5. Taylor expanded in kx around 0 14.3%

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

   if -4.50000000000000013e-19 < ky < 1.12e-122

   1. Initial program 80.7%

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

      1. associate-*l/ 79.6%

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

      2. +-commutative 79.6%

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

      3. unpow2 79.6%

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

      4. unpow2 79.6%

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

      5. hypot-def 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in th around 0 32.7%

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

   5. Taylor expanded in ky around 0 20.5%

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

      1. associate-/l* 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in kx around 0 20.3%

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

      1. associate-/l* 21.3%

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

   10. Simplified 21.3%

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

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq -4.5 \cdot 10^{-19}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.12 \cdot 10^{-122}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 19: 13.9% accurate, 709.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.7%

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

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

   3. unpow2 92.2%

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

   4. unpow2 92.2%

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

   5. hypot-def 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in th around 0 43.2%

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

5. Taylor expanded in kx around 0 11.1%

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

6. Final simplification 11.1%

   \[\leadsto th \]

Reproduce

herbie shell --seed 2023189 
(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)))