Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.5% → 99.6%

Time: 12.6s

Alternatives: 20

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.5% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky\_m}{\mathsf{hypot}\left(\sin ky\_m, \sin kx\right)}\right)\right) \cdot \sin th\right) \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (* (expm1 (log1p (/ (sin ky_m) (hypot (sin ky_m) (sin kx))))) (sin th))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * (expm1(log1p((sin(ky_m) / hypot(sin(ky_m), sin(kx))))) * sin(th));
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * (Math.expm1(Math.log1p((Math.sin(ky_m) / Math.hypot(Math.sin(ky_m), Math.sin(kx))))) * Math.sin(th));
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	return ky_s * (math.expm1(math.log1p((math.sin(ky_m) / math.hypot(math.sin(ky_m), math.sin(kx))))) * math.sin(th))

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	return Float64(ky_s * Float64(expm1(log1p(Float64(sin(ky_m) / hypot(sin(ky_m), sin(kx))))) * sin(th)))
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * N[(N[(Exp[N[Log[1 + N[(N[Sin[ky$95$m], $MachinePrecision] / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. expm1-log1p-u 91.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)} \cdot \sin th \]

   2. +-commutative 91.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}\right)\right) \cdot \sin th \]

   3. unpow2 91.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}\right)\right) \cdot \sin th \]

   4. unpow2 91.4%

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

   5. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 76.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq -0.02:\\ \;\;\;\;\sin ky\_m \cdot \frac{th}{\mathsf{hypot}\left(\sin ky\_m, \sin kx\right)}\\ \mathbf{elif}\;\sin ky\_m \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin ky\_m \cdot \frac{\sin th}{\mathsf{hypot}\left(ky\_m, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) -0.02)
    (* (sin ky_m) (/ th (hypot (sin ky_m) (sin kx))))
    (if (<= (sin ky_m) 2e-7)
      (* (sin ky_m) (/ (sin th) (hypot ky_m (sin kx))))
      (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= -0.02) {
		tmp = sin(ky_m) * (th / hypot(sin(ky_m), sin(kx)));
	} else if (sin(ky_m) <= 2e-7) {
		tmp = sin(ky_m) * (sin(th) / hypot(ky_m, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= -0.02) {
		tmp = Math.sin(ky_m) * (th / Math.hypot(Math.sin(ky_m), Math.sin(kx)));
	} else if (Math.sin(ky_m) <= 2e-7) {
		tmp = Math.sin(ky_m) * (Math.sin(th) / Math.hypot(ky_m, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= -0.02:
		tmp = math.sin(ky_m) * (th / math.hypot(math.sin(ky_m), math.sin(kx)))
	elif math.sin(ky_m) <= 2e-7:
		tmp = math.sin(ky_m) * (math.sin(th) / math.hypot(ky_m, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= -0.02)
		tmp = Float64(sin(ky_m) * Float64(th / hypot(sin(ky_m), sin(kx))));
	elseif (sin(ky_m) <= 2e-7)
		tmp = Float64(sin(ky_m) * Float64(sin(th) / hypot(ky_m, sin(kx))));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= -0.02)
		tmp = sin(ky_m) * (th / hypot(sin(ky_m), sin(kx)));
	elseif (sin(ky_m) <= 2e-7)
		tmp = sin(ky_m) * (sin(th) / hypot(ky_m, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], -0.02], N[(N[Sin[ky$95$m], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 2e-7], N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky$95$m ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 50.6%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 1.9999999999999999e-7

   1. Initial program 84.4%

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

      1. unpow2 84.4%

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

      2. sqr-neg 84.4%

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

      3. sin-neg 84.4%

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

      4. sin-neg 84.4%

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

      5. unpow2 84.4%

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

      6. associate-*l/ 82.0%

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

      7. associate-/l* 84.3%

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

      8. +-commutative 84.3%

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

      9. unpow2 84.3%

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

      10. sin-neg 84.3%

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

      11. sin-neg 84.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ky around 0 99.2%

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

   if 1.9999999999999999e-7 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 66.4%

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

Alternative 3: 57.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq -0.02:\\ \;\;\;\;\frac{\sin ky\_m}{\frac{\mathsf{hypot}\left(\sin ky\_m, kx\right)}{th}}\\ \mathbf{elif}\;\sin ky\_m \leq 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky\_m}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) -0.02)
    (/ (sin ky_m) (/ (hypot (sin ky_m) kx) th))
    (if (<= (sin ky_m) 1e-94)
      (* (sin th) (/ (sin ky_m) (fabs (sin kx))))
      (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= -0.02) {
		tmp = sin(ky_m) / (hypot(sin(ky_m), kx) / th);
	} else if (sin(ky_m) <= 1e-94) {
		tmp = sin(th) * (sin(ky_m) / fabs(sin(kx)));
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= -0.02) {
		tmp = Math.sin(ky_m) / (Math.hypot(Math.sin(ky_m), kx) / th);
	} else if (Math.sin(ky_m) <= 1e-94) {
		tmp = Math.sin(th) * (Math.sin(ky_m) / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= -0.02:
		tmp = math.sin(ky_m) / (math.hypot(math.sin(ky_m), kx) / th)
	elif math.sin(ky_m) <= 1e-94:
		tmp = math.sin(th) * (math.sin(ky_m) / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= -0.02)
		tmp = Float64(sin(ky_m) / Float64(hypot(sin(ky_m), kx) / th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = Float64(sin(th) * Float64(sin(ky_m) / abs(sin(kx))));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= -0.02)
		tmp = sin(ky_m) / (hypot(sin(ky_m), kx) / th);
	elseif (sin(ky_m) <= 1e-94)
		tmp = sin(th) * (sin(ky_m) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], -0.02], N[(N[Sin[ky$95$m], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 1e-94], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky$95$m], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 50.6%

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

   8. Taylor expanded in kx around 0 33.1%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.9999999999999996e-95

   1. Initial program 82.6%

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

   3. Taylor expanded in ky around 0 44.6%

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

      1. add-sqr-sqrt 39.0%

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

      2. sqrt-prod 62.9%

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

      3. rem-sqrt-square 74.3%

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

   5. Applied egg-rr 74.3%

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

   if 9.9999999999999996e-95 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 64.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, kx\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq -0.02:\\ \;\;\;\;\frac{\sin ky\_m}{\frac{\mathsf{hypot}\left(\sin ky\_m, kx\right)}{th}}\\ \mathbf{elif}\;\sin ky\_m \leq 10^{-94}:\\ \;\;\;\;\frac{1}{\frac{\frac{\left|\sin kx\right|}{\sin th}}{ky\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) -0.02)
    (/ (sin ky_m) (/ (hypot (sin ky_m) kx) th))
    (if (<= (sin ky_m) 1e-94)
      (/ 1.0 (/ (/ (fabs (sin kx)) (sin th)) ky_m))
      (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= -0.02) {
		tmp = sin(ky_m) / (hypot(sin(ky_m), kx) / th);
	} else if (sin(ky_m) <= 1e-94) {
		tmp = 1.0 / ((fabs(sin(kx)) / sin(th)) / ky_m);
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= -0.02) {
		tmp = Math.sin(ky_m) / (Math.hypot(Math.sin(ky_m), kx) / th);
	} else if (Math.sin(ky_m) <= 1e-94) {
		tmp = 1.0 / ((Math.abs(Math.sin(kx)) / Math.sin(th)) / ky_m);
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= -0.02:
		tmp = math.sin(ky_m) / (math.hypot(math.sin(ky_m), kx) / th)
	elif math.sin(ky_m) <= 1e-94:
		tmp = 1.0 / ((math.fabs(math.sin(kx)) / math.sin(th)) / ky_m)
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= -0.02)
		tmp = Float64(sin(ky_m) / Float64(hypot(sin(ky_m), kx) / th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = Float64(1.0 / Float64(Float64(abs(sin(kx)) / sin(th)) / ky_m));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= -0.02)
		tmp = sin(ky_m) / (hypot(sin(ky_m), kx) / th);
	elseif (sin(ky_m) <= 1e-94)
		tmp = 1.0 / ((abs(sin(kx)) / sin(th)) / ky_m);
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], -0.02], N[(N[Sin[ky$95$m], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 1e-94], N[(1.0 / N[(N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] / ky$95$m), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 50.6%

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

   8. Taylor expanded in kx around 0 33.1%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.9999999999999996e-95

   1. Initial program 82.6%

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

   3. Taylor expanded in ky around 0 44.6%

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

      1. associate-*l/ 42.2%

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

      2. clear-num 41.7%

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

   5. Applied egg-rr 41.7%

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

   6. Taylor expanded in ky around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/r* 43.9%

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

   8. Simplified 43.9%

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

      1. add-sqr-sqrt 39.0%

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

      2. sqrt-prod 62.9%

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

      3. rem-sqrt-square 74.3%

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

   10. Applied egg-rr 73.3%

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

   if 9.9999999999999996e-95 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 64.9%

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

Alternative 5: 46.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sin ky\_m}{\frac{\mathsf{hypot}\left(\sin ky\_m, kx\right)}{th}}\\ \mathbf{elif}\;\sin ky\_m \leq 10^{-94}:\\ \;\;\;\;\left|ky\_m \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) 5e-308)
    (/ (sin ky_m) (/ (hypot (sin ky_m) kx) th))
    (if (<= (sin ky_m) 1e-94)
      (fabs (* ky_m (/ (sin th) (sin kx))))
      (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= 5e-308) {
		tmp = sin(ky_m) / (hypot(sin(ky_m), kx) / th);
	} else if (sin(ky_m) <= 1e-94) {
		tmp = fabs((ky_m * (sin(th) / sin(kx))));
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= 5e-308) {
		tmp = Math.sin(ky_m) / (Math.hypot(Math.sin(ky_m), kx) / th);
	} else if (Math.sin(ky_m) <= 1e-94) {
		tmp = Math.abs((ky_m * (Math.sin(th) / Math.sin(kx))));
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= 5e-308:
		tmp = math.sin(ky_m) / (math.hypot(math.sin(ky_m), kx) / th)
	elif math.sin(ky_m) <= 1e-94:
		tmp = math.fabs((ky_m * (math.sin(th) / math.sin(kx))))
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= 5e-308)
		tmp = Float64(sin(ky_m) / Float64(hypot(sin(ky_m), kx) / th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = abs(Float64(ky_m * Float64(sin(th) / sin(kx))));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= 5e-308)
		tmp = sin(ky_m) / (hypot(sin(ky_m), kx) / th);
	elseif (sin(ky_m) <= 1e-94)
		tmp = abs((ky_m * (sin(th) / sin(kx))));
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 5e-308], N[(N[Sin[ky$95$m], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 1e-94], N[Abs[N[(ky$95$m * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 4.99999999999999955e-308

   1. Initial program 91.6%

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

      1. unpow2 91.6%

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

      2. sqr-neg 91.6%

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

      3. sin-neg 91.6%

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

      4. sin-neg 91.6%

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

      5. unpow2 91.6%

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

      6. associate-*l/ 89.1%

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

      7. associate-/l* 91.5%

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

      8. +-commutative 91.5%

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

      9. unpow2 91.5%

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

      10. sin-neg 91.5%

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

      11. sin-neg 91.5%

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

   3. Simplified 99.6%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 51.8%

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

   8. Taylor expanded in kx around 0 37.0%

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

   if 4.99999999999999955e-308 < (sin.f64 ky) < 9.9999999999999996e-95

   1. Initial program 79.3%

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

   3. Taylor expanded in ky around 0 51.6%

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

      1. associate-*l/ 51.6%

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

      2. clear-num 50.5%

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

   5. Applied egg-rr 50.5%

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

   6. Taylor expanded in ky around 0 50.5%

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

      1. *-commutative 50.5%

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

      2. associate-/r* 50.4%

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

   8. Simplified 50.4%

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

      1. add-sqr-sqrt 29.7%

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

      2. sqrt-unprod 37.8%

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

      3. pow2 37.8%

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

      4. clear-num 37.9%

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

      5. associate-/r/ 37.9%

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

   10. Applied egg-rr 37.9%

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

       1. unpow2 37.9%

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

       2. rem-sqrt-square 51.0%

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

       3. associate-*l/ 49.1%

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

       4. associate-*r/ 50.8%

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

   12. Simplified 50.8%

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

   if 9.9999999999999996e-95 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 64.9%

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

Alternative 6: 48.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky\_m \leq 10^{-94}:\\ \;\;\;\;\left|ky\_m \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) -0.02)
    (fabs (sin th))
    (if (<= (sin ky_m) 1e-94)
      (fabs (* ky_m (/ (sin th) (sin kx))))
      (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky_m) <= 1e-94) {
		tmp = fabs((ky_m * (sin(th) / sin(kx))));
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky_m) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky_m) <= 1d-94) then
        tmp = abs((ky_m * (sin(th) / sin(kx))))
    else
        tmp = sin(th)
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky_m) <= 1e-94) {
		tmp = Math.abs((ky_m * (Math.sin(th) / Math.sin(kx))));
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky_m) <= 1e-94:
		tmp = math.fabs((ky_m * (math.sin(th) / math.sin(kx))))
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = abs(Float64(ky_m * Float64(sin(th) / sin(kx))));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = abs((ky_m * (sin(th) / sin(kx))));
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 1e-94], N[Abs[N[(ky$95$m * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

   7. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 31.7%

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

   9. Simplified 31.7%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.9999999999999996e-95

   1. Initial program 82.6%

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

   3. Taylor expanded in ky around 0 44.6%

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

      1. associate-*l/ 42.2%

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

      2. clear-num 41.7%

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

   5. Applied egg-rr 41.7%

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

   6. Taylor expanded in ky around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/r* 43.9%

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

   8. Simplified 43.9%

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

      1. add-sqr-sqrt 25.0%

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

      2. sqrt-unprod 29.0%

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

      3. pow2 29.0%

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

      4. clear-num 29.1%

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

      5. associate-/r/ 29.1%

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

   10. Applied egg-rr 29.1%

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

       1. unpow2 29.1%

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

       2. rem-sqrt-square 40.7%

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

       3. associate-*l/ 39.0%

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

       4. associate-*r/ 40.6%

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

   12. Simplified 40.6%

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

   if 9.9999999999999996e-95 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 64.9%

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

Alternative 7: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \left(\frac{\sin ky\_m}{\mathsf{hypot}\left(\sin ky\_m, \sin kx\right)} \cdot \sin th\right) \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (* (/ (sin ky_m) (hypot (sin ky_m) (sin kx))) (sin th))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * ((sin(ky_m) / hypot(sin(ky_m), sin(kx))) * sin(th));
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * ((Math.sin(ky_m) / Math.hypot(Math.sin(ky_m), Math.sin(kx))) * Math.sin(th));
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	return ky_s * ((math.sin(ky_m) / math.hypot(math.sin(ky_m), math.sin(kx))) * math.sin(th))

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	return Float64(ky_s * Float64(Float64(sin(ky_m) / hypot(sin(ky_m), sin(kx))) * sin(th)))
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp = code(ky_s, kx, ky_m, th)
	tmp = ky_s * ((sin(ky_m) / hypot(sin(ky_m), sin(kx))) * sin(th));
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * N[(N[(N[Sin[ky$95$m], $MachinePrecision] / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. +-commutative 91.5%

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

   2. unpow2 91.5%

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

   3. unpow2 91.5%

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

   4. hypot-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 8: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \left(\sin ky\_m \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky\_m, \sin kx\right)}\right) \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (* (sin ky_m) (/ (sin th) (hypot (sin ky_m) (sin kx))))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * (sin(ky_m) * (sin(th) / hypot(sin(ky_m), sin(kx))));
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * (Math.sin(ky_m) * (Math.sin(th) / Math.hypot(Math.sin(ky_m), Math.sin(kx))));
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	return ky_s * (math.sin(ky_m) * (math.sin(th) / math.hypot(math.sin(ky_m), math.sin(kx))))

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	return Float64(ky_s * Float64(sin(ky_m) * Float64(sin(th) / hypot(sin(ky_m), sin(kx)))))
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp = code(ky_s, kx, ky_m, th)
	tmp = ky_s * (sin(ky_m) * (sin(th) / hypot(sin(ky_m), sin(kx))));
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. unpow2 91.5%

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

   2. sqr-neg 91.5%

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

   3. sin-neg 91.5%

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

   4. sin-neg 91.5%

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

   5. unpow2 91.5%

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

   6. associate-*l/ 90.2%

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

   7. associate-/l* 91.4%

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

   8. +-commutative 91.4%

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

   9. unpow2 91.4%

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

   10. sin-neg 91.4%

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

   11. sin-neg 91.4%

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

3. Simplified 99.6%

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

Alternative 9: 59.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;th \leq 0.0077:\\ \;\;\;\;\sin ky\_m \cdot \frac{th}{\mathsf{hypot}\left(\sin ky\_m, \sin kx\right)}\\ \mathbf{elif}\;th \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky\_m}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= th 0.0077)
    (* (sin ky_m) (/ th (hypot (sin ky_m) (sin kx))))
    (if (<= th 3e+84) (sin th) (* (sin th) (/ (sin ky_m) (fabs (sin kx))))))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (th <= 0.0077) {
		tmp = sin(ky_m) * (th / hypot(sin(ky_m), sin(kx)));
	} else if (th <= 3e+84) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (sin(ky_m) / fabs(sin(kx)));
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (th <= 0.0077) {
		tmp = Math.sin(ky_m) * (th / Math.hypot(Math.sin(ky_m), Math.sin(kx)));
	} else if (th <= 3e+84) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.sin(ky_m) / Math.abs(Math.sin(kx)));
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if th <= 0.0077:
		tmp = math.sin(ky_m) * (th / math.hypot(math.sin(ky_m), math.sin(kx)))
	elif th <= 3e+84:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (math.sin(ky_m) / math.fabs(math.sin(kx)))
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (th <= 0.0077)
		tmp = Float64(sin(ky_m) * Float64(th / hypot(sin(ky_m), sin(kx))));
	elseif (th <= 3e+84)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(sin(ky_m) / abs(sin(kx))));
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (th <= 0.0077)
		tmp = sin(ky_m) * (th / hypot(sin(ky_m), sin(kx)));
	elseif (th <= 3e+84)
		tmp = sin(th);
	else
		tmp = sin(th) * (sin(ky_m) / abs(sin(kx)));
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[th, 0.0077], N[(N[Sin[ky$95$m], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3e+84], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky$95$m], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if th < 0.0077000000000000002

   1. Initial program 91.2%

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

      1. unpow2 91.2%

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

      2. sqr-neg 91.2%

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

      3. sin-neg 91.2%

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

      4. sin-neg 91.2%

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

      5. unpow2 91.2%

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

      6. associate-*l/ 89.6%

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

      7. associate-/l* 91.1%

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

      8. +-commutative 91.1%

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

      9. unpow2 91.1%

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

      10. sin-neg 91.1%

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

      11. sin-neg 91.1%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 64.5%

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

   if 0.0077000000000000002 < th < 2.99999999999999996e84

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.0%

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

      8. +-commutative 99.0%

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

      9. unpow2 99.0%

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

      10. sin-neg 99.0%

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

      11. sin-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in kx around 0 4.0%

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

   if 2.99999999999999996e84 < th

   1. Initial program 91.2%

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

   3. Taylor expanded in ky around 0 21.1%

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

      1. add-sqr-sqrt 19.6%

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

      2. sqrt-prod 43.3%

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

      3. rem-sqrt-square 49.9%

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

   5. Applied egg-rr 49.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 0.0077:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;th \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky\_m \leq 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky\_m}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) -0.02)
    (fabs (sin th))
    (if (<= (sin ky_m) 1e-94) (* (sin th) (/ ky_m (sin kx))) (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky_m) <= 1e-94) {
		tmp = sin(th) * (ky_m / sin(kx));
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky_m) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky_m) <= 1d-94) then
        tmp = sin(th) * (ky_m / sin(kx))
    else
        tmp = sin(th)
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky_m) <= 1e-94) {
		tmp = Math.sin(th) * (ky_m / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky_m) <= 1e-94:
		tmp = math.sin(th) * (ky_m / math.sin(kx))
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = Float64(sin(th) * Float64(ky_m / sin(kx)));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = sin(th) * (ky_m / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 1e-94], N[(N[Sin[th], $MachinePrecision] * N[(ky$95$m / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

   7. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 31.7%

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

   9. Simplified 31.7%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.9999999999999996e-95

   1. Initial program 82.6%

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

   3. Taylor expanded in ky around 0 44.6%

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

   if 9.9999999999999996e-95 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 64.9%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;\sin ky\_m \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky\_m \leq 10^{-94}:\\ \;\;\;\;ky\_m \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= (sin ky_m) -0.02)
    (fabs (sin th))
    (if (<= (sin ky_m) 1e-94) (* ky_m (/ (sin th) (sin kx))) (sin th)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (sin(ky_m) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky_m) <= 1e-94) {
		tmp = ky_m * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky_m) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky_m) <= 1d-94) then
        tmp = ky_m * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (Math.sin(ky_m) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky_m) <= 1e-94) {
		tmp = ky_m * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if math.sin(ky_m) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky_m) <= 1e-94:
		tmp = ky_m * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (sin(ky_m) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = Float64(ky_m * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (sin(ky_m) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky_m) <= 1e-94)
		tmp = ky_m * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky$95$m], $MachinePrecision], 1e-94], N[(ky$95$m * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 2.6%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 25.1%

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

      3. pow2 25.1%

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

   7. Applied egg-rr 25.1%

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

      1. unpow2 25.1%

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

      2. rem-sqrt-square 31.7%

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

   9. Simplified 31.7%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.9999999999999996e-95

   1. Initial program 82.6%

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

      1. unpow2 82.6%

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

      2. sqr-neg 82.6%

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

      3. sin-neg 82.6%

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

      4. sin-neg 82.6%

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

      5. unpow2 82.6%

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

      6. associate-*l/ 80.0%

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

      7. associate-/l* 82.6%

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

      8. +-commutative 82.6%

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

      9. unpow2 82.6%

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

      10. sin-neg 82.6%

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

      11. sin-neg 82.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ky around 0 42.2%

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

      1. associate-/l* 44.6%

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

   7. Simplified 44.6%

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

   if 9.9999999999999996e-95 < (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. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 64.9%

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

Alternative 12: 68.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;kx \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky\_m}{\mathsf{hypot}\left(\sin ky\_m, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky\_m}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= kx 2.1e-6)
    (* (sin th) (/ (sin ky_m) (hypot (sin ky_m) kx)))
    (* (sin th) (/ (sin ky_m) (fabs (sin kx)))))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 2.1e-6) {
		tmp = sin(th) * (sin(ky_m) / hypot(sin(ky_m), kx));
	} else {
		tmp = sin(th) * (sin(ky_m) / fabs(sin(kx)));
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 2.1e-6) {
		tmp = Math.sin(th) * (Math.sin(ky_m) / Math.hypot(Math.sin(ky_m), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky_m) / Math.abs(Math.sin(kx)));
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if kx <= 2.1e-6:
		tmp = math.sin(th) * (math.sin(ky_m) / math.hypot(math.sin(ky_m), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky_m) / math.fabs(math.sin(kx)))
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (kx <= 2.1e-6)
		tmp = Float64(sin(th) * Float64(sin(ky_m) / hypot(sin(ky_m), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky_m) / abs(sin(kx))));
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (kx <= 2.1e-6)
		tmp = sin(th) * (sin(ky_m) / hypot(sin(ky_m), kx));
	else
		tmp = sin(th) * (sin(ky_m) / abs(sin(kx)));
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[kx, 2.1e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky$95$m], $MachinePrecision] / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky$95$m], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if kx < 2.0999999999999998e-6

   1. Initial program 89.5%

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

      1. +-commutative 89.5%

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

      2. unpow2 89.5%

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

      3. unpow2 89.5%

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

      4. hypot-undefine 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in kx around 0 73.9%

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

   if 2.0999999999999998e-6 < kx

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 33.6%

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

      1. add-sqr-sqrt 24.1%

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

      2. sqrt-prod 61.3%

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

      3. rem-sqrt-square 61.3%

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

   5. Applied egg-rr 61.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;kx \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\sin ky\_m \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky\_m, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky\_m}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= kx 2.1e-6)
    (* (sin ky_m) (/ (sin th) (hypot (sin ky_m) kx)))
    (* (sin th) (/ (sin ky_m) (fabs (sin kx)))))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 2.1e-6) {
		tmp = sin(ky_m) * (sin(th) / hypot(sin(ky_m), kx));
	} else {
		tmp = sin(th) * (sin(ky_m) / fabs(sin(kx)));
	}
	return ky_s * tmp;
}

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 2.1e-6) {
		tmp = Math.sin(ky_m) * (Math.sin(th) / Math.hypot(Math.sin(ky_m), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky_m) / Math.abs(Math.sin(kx)));
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if kx <= 2.1e-6:
		tmp = math.sin(ky_m) * (math.sin(th) / math.hypot(math.sin(ky_m), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky_m) / math.fabs(math.sin(kx)))
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (kx <= 2.1e-6)
		tmp = Float64(sin(ky_m) * Float64(sin(th) / hypot(sin(ky_m), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky_m) / abs(sin(kx))));
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (kx <= 2.1e-6)
		tmp = sin(ky_m) * (sin(th) / hypot(sin(ky_m), kx));
	else
		tmp = sin(th) * (sin(ky_m) / abs(sin(kx)));
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[kx, 2.1e-6], N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky$95$m], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if kx < 2.0999999999999998e-6

   1. Initial program 89.5%

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

      1. unpow2 89.5%

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

      2. sqr-neg 89.5%

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

      3. sin-neg 89.5%

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

      4. sin-neg 89.5%

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

      5. unpow2 89.5%

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

      6. associate-*l/ 87.9%

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

      7. associate-/l* 89.4%

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

      8. +-commutative 89.4%

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

      9. unpow2 89.4%

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

      10. sin-neg 89.4%

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

      11. sin-neg 89.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 73.8%

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

   if 2.0999999999999998e-6 < kx

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 33.6%

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

      1. add-sqr-sqrt 24.1%

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

      2. sqrt-prod 61.3%

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

      3. rem-sqrt-square 61.3%

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

   5. Applied egg-rr 61.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left|\sin kx\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;ky\_m \leq 4.5 \cdot 10^{-146}:\\ \;\;\;\;ky\_m \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;ky\_m \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\left|\sin th\right|\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (*
  ky_s
  (if (<= ky_m 4.5e-146)
    (* ky_m (/ (sin th) kx))
    (if (<= ky_m 2.5e+107) (sin th) (fabs (sin th))))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 4.5e-146) {
		tmp = ky_m * (sin(th) / kx);
	} else if (ky_m <= 2.5e+107) {
		tmp = sin(th);
	} else {
		tmp = fabs(sin(th));
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky_m <= 4.5d-146) then
        tmp = ky_m * (sin(th) / kx)
    else if (ky_m <= 2.5d+107) then
        tmp = sin(th)
    else
        tmp = abs(sin(th))
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 4.5e-146) {
		tmp = ky_m * (Math.sin(th) / kx);
	} else if (ky_m <= 2.5e+107) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.abs(Math.sin(th));
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if ky_m <= 4.5e-146:
		tmp = ky_m * (math.sin(th) / kx)
	elif ky_m <= 2.5e+107:
		tmp = math.sin(th)
	else:
		tmp = math.fabs(math.sin(th))
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (ky_m <= 4.5e-146)
		tmp = Float64(ky_m * Float64(sin(th) / kx));
	elseif (ky_m <= 2.5e+107)
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (ky_m <= 4.5e-146)
		tmp = ky_m * (sin(th) / kx);
	elseif (ky_m <= 2.5e+107)
		tmp = sin(th);
	else
		tmp = abs(sin(th));
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[ky$95$m, 4.5e-146], N[(ky$95$m * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky$95$m, 2.5e+107], N[Sin[th], $MachinePrecision], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if ky < 4.5000000000000001e-146

   1. Initial program 87.7%

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

   3. Taylor expanded in ky around 0 30.3%

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

      1. associate-*l/ 28.6%

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

      2. clear-num 28.3%

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

   5. Applied egg-rr 28.3%

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

   6. Taylor expanded in ky around 0 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r* 28.4%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in kx around 0 20.7%

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

       1. associate-/l* 22.5%

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

   11. Simplified 22.5%

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

   if 4.5000000000000001e-146 < ky < 2.5000000000000001e107

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.4%

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

      8. +-commutative 99.4%

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

      9. unpow2 99.4%

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

      10. sin-neg 99.4%

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

      11. sin-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in kx around 0 44.3%

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

   if 2.5000000000000001e107 < ky

   1. Initial program 99.8%

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

      1. unpow2 99.8%

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

      2. sqr-neg 99.8%

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

      3. sin-neg 99.8%

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

      4. sin-neg 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. sin-neg 99.6%

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

      11. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 31.0%

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

      1. add-sqr-sqrt 6.6%

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

      2. sqrt-unprod 18.5%

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

      3. pow2 18.5%

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

   7. Applied egg-rr 18.5%

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

      1. unpow2 18.5%

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

      2. rem-sqrt-square 19.5%

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

   9. Simplified 19.5%

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

Alternative 15: 36.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;ky\_m \leq 3.85 \cdot 10^{-146}:\\ \;\;\;\;ky\_m \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (if (<= ky_m 3.85e-146) (* ky_m (/ (sin th) kx)) (sin th))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 3.85e-146) {
		tmp = ky_m * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky_m <= 3.85d-146) then
        tmp = ky_m * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 3.85e-146) {
		tmp = ky_m * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if ky_m <= 3.85e-146:
		tmp = ky_m * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (ky_m <= 3.85e-146)
		tmp = Float64(ky_m * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (ky_m <= 3.85e-146)
		tmp = ky_m * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[ky$95$m, 3.85e-146], N[(ky$95$m * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if ky < 3.84999999999999998e-146

   1. Initial program 87.7%

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

   3. Taylor expanded in ky around 0 30.3%

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

      1. associate-*l/ 28.6%

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

      2. clear-num 28.3%

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

   5. Applied egg-rr 28.3%

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

   6. Taylor expanded in ky around 0 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r* 28.4%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in kx around 0 20.7%

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

       1. associate-/l* 22.5%

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

   11. Simplified 22.5%

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

   if 3.84999999999999998e-146 < ky

   1. Initial program 99.7%

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

      1. unpow2 99.7%

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

      2. sqr-neg 99.7%

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

      3. sin-neg 99.7%

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

      4. sin-neg 99.7%

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

      5. unpow2 99.7%

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

      6. associate-*l/ 99.6%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 39.5%

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

Alternative 16: 33.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;kx \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;ky\_m \cdot \frac{th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (if (<= kx 1.95e-7) (sin th) (* ky_m (/ th (sin kx))))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 1.95e-7) {
		tmp = sin(th);
	} else {
		tmp = ky_m * (th / sin(kx));
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.95d-7) then
        tmp = sin(th)
    else
        tmp = ky_m * (th / sin(kx))
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 1.95e-7) {
		tmp = Math.sin(th);
	} else {
		tmp = ky_m * (th / Math.sin(kx));
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if kx <= 1.95e-7:
		tmp = math.sin(th)
	else:
		tmp = ky_m * (th / math.sin(kx))
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (kx <= 1.95e-7)
		tmp = sin(th);
	else
		tmp = Float64(ky_m * Float64(th / sin(kx)));
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (kx <= 1.95e-7)
		tmp = sin(th);
	else
		tmp = ky_m * (th / sin(kx));
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[kx, 1.95e-7], N[Sin[th], $MachinePrecision], N[(ky$95$m * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if kx < 1.95000000000000012e-7

   1. Initial program 89.5%

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

      1. unpow2 89.5%

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

      2. sqr-neg 89.5%

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

      3. sin-neg 89.5%

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

      4. sin-neg 89.5%

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

      5. unpow2 89.5%

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

      6. associate-*l/ 87.9%

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

      7. associate-/l* 89.4%

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

      8. +-commutative 89.4%

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

      9. unpow2 89.4%

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

      10. sin-neg 89.4%

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

      11. sin-neg 89.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.9%

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

   if 1.95000000000000012e-7 < kx

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0 33.6%

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

      1. associate-*l/ 33.6%

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

      2. clear-num 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in ky around 0 29.2%

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

      1. *-commutative 29.2%

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

      2. associate-/r* 29.2%

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

   8. Simplified 29.2%

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

   9. Taylor expanded in th around 0 22.2%

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

       1. associate-/l* 22.2%

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

   11. Simplified 22.2%

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

Alternative 17: 31.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;kx \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\log \left(th + 1\right)\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (if (<= kx 1.2e-7) (sin th) (log (+ th 1.0)))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 1.2e-7) {
		tmp = sin(th);
	} else {
		tmp = log((th + 1.0));
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.2d-7) then
        tmp = sin(th)
    else
        tmp = log((th + 1.0d0))
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (kx <= 1.2e-7) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.log((th + 1.0));
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if kx <= 1.2e-7:
		tmp = math.sin(th)
	else:
		tmp = math.log((th + 1.0))
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (kx <= 1.2e-7)
		tmp = sin(th);
	else
		tmp = log(Float64(th + 1.0));
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (kx <= 1.2e-7)
		tmp = sin(th);
	else
		tmp = log((th + 1.0));
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[kx, 1.2e-7], N[Sin[th], $MachinePrecision], N[Log[N[(th + 1.0), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if kx < 1.19999999999999989e-7

   1. Initial program 89.5%

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

      1. unpow2 89.5%

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

      2. sqr-neg 89.5%

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

      3. sin-neg 89.5%

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

      4. sin-neg 89.5%

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

      5. unpow2 89.5%

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

      6. associate-*l/ 87.9%

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

      7. associate-/l* 89.4%

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

      8. +-commutative 89.4%

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

      9. unpow2 89.4%

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

      10. sin-neg 89.4%

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

      11. sin-neg 89.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 27.9%

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

   if 1.19999999999999989e-7 < kx

   1. Initial program 99.6%

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

      1. unpow2 99.6%

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

      2. sqr-neg 99.6%

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

      3. sin-neg 99.6%

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

      4. sin-neg 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l/ 99.5%

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

      7. associate-/l* 99.5%

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

      8. +-commutative 99.5%

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

      9. unpow2 99.5%

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

      10. sin-neg 99.5%

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

      11. sin-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 7.8%

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

      1. add-log-exp 19.7%

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

   7. Applied egg-rr 19.7%

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

   8. Taylor expanded in th around 0 18.0%

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

      1. +-commutative 18.0%

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

   10. Simplified 18.0%

       \[\leadsto \log \color{blue}{\left(th + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 32.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;ky\_m \leq 1.95 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (if (<= ky_m 1.95e-234) 0.0 (sin th))))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 1.95e-234) {
		tmp = 0.0;
	} else {
		tmp = sin(th);
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky_m <= 1.95d-234) then
        tmp = 0.0d0
    else
        tmp = sin(th)
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 1.95e-234) {
		tmp = 0.0;
	} else {
		tmp = Math.sin(th);
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if ky_m <= 1.95e-234:
		tmp = 0.0
	else:
		tmp = math.sin(th)
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (ky_m <= 1.95e-234)
		tmp = 0.0;
	else
		tmp = sin(th);
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (ky_m <= 1.95e-234)
		tmp = 0.0;
	else
		tmp = sin(th);
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[ky$95$m, 1.95e-234], 0.0, N[Sin[th], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if ky < 1.9500000000000001e-234

   1. Initial program 90.0%

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

      1. unpow2 90.0%

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

      2. sqr-neg 90.0%

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

      3. sin-neg 90.0%

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

      4. sin-neg 90.0%

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

      5. unpow2 90.0%

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

      6. associate-*l/ 87.9%

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

      7. associate-/l* 89.9%

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

      8. +-commutative 89.9%

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

      9. unpow2 89.9%

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

      10. sin-neg 89.9%

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

      11. sin-neg 89.9%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in kx around 0 15.8%

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

      1. add-log-exp 20.1%

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

   7. Applied egg-rr 20.1%

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

   8. Taylor expanded in th around 0 15.3%

      \[\leadsto \log \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 15.3%

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 15.3%

       \[\leadsto \color{blue}{0} \]

   if 1.9500000000000001e-234 < ky

   1. Initial program 93.8%

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

      1. unpow2 93.8%

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

      2. sqr-neg 93.8%

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

      3. sin-neg 93.8%

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

      4. sin-neg 93.8%

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

      5. unpow2 93.8%

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

      6. associate-*l/ 93.6%

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

      7. associate-/l* 93.5%

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

      8. +-commutative 93.5%

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

      9. unpow2 93.5%

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

      10. sin-neg 93.5%

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

      11. sin-neg 93.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 35.4%

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

Alternative 19: 21.4% accurate, 117.9× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot \begin{array}{l} \mathbf{if}\;ky\_m \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th)
 :precision binary64
 (* ky_s (if (<= ky_m 4.6e-183) 0.0 th)))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 4.6e-183) {
		tmp = 0.0;
	} else {
		tmp = th;
	}
	return ky_s * tmp;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky_m <= 4.6d-183) then
        tmp = 0.0d0
    else
        tmp = th
    end if
    code = ky_s * tmp
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	double tmp;
	if (ky_m <= 4.6e-183) {
		tmp = 0.0;
	} else {
		tmp = th;
	}
	return ky_s * tmp;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	tmp = 0
	if ky_m <= 4.6e-183:
		tmp = 0.0
	else:
		tmp = th
	return ky_s * tmp

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	tmp = 0.0
	if (ky_m <= 4.6e-183)
		tmp = 0.0;
	else
		tmp = th;
	end
	return Float64(ky_s * tmp)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp_2 = code(ky_s, kx, ky_m, th)
	tmp = 0.0;
	if (ky_m <= 4.6e-183)
		tmp = 0.0;
	else
		tmp = th;
	end
	tmp_2 = ky_s * tmp;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * If[LessEqual[ky$95$m, 4.6e-183], 0.0, th]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if ky < 4.60000000000000032e-183

   1. Initial program 88.4%

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

      1. unpow2 88.4%

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

      2. sqr-neg 88.4%

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

      3. sin-neg 88.4%

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

      4. sin-neg 88.4%

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

      5. unpow2 88.4%

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

      6. associate-*l/ 86.4%

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

      7. associate-/l* 88.3%

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

      8. +-commutative 88.3%

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

      9. unpow2 88.3%

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

      10. sin-neg 88.3%

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

      11. sin-neg 88.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 16.1%

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

      1. add-log-exp 20.3%

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

   7. Applied egg-rr 20.3%

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

   8. Taylor expanded in th around 0 14.7%

      \[\leadsto \log \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 14.7%

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 14.7%

       \[\leadsto \color{blue}{0} \]

   if 4.60000000000000032e-183 < ky

   1. Initial program 97.1%

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

      1. unpow2 97.1%

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

      2. sqr-neg 97.1%

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

      3. sin-neg 97.1%

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

      4. sin-neg 97.1%

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

      5. unpow2 97.1%

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

      6. associate-*l/ 97.0%

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

      7. associate-/l* 96.9%

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

      8. +-commutative 96.9%

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

      9. unpow2 96.9%

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

      10. sin-neg 96.9%

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

      11. sin-neg 96.9%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in kx around 0 37.5%

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

   6. Taylor expanded in th around 0 21.1%

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

Alternative 20: 12.4% accurate, 709.0× speedup

Math

\[\begin{array}{l} ky\_m = \left|ky\right| \\ ky\_s = \mathsf{copysign}\left(1, ky\right) \\ ky\_s \cdot 0 \end{array} \]

FPCore

ky\_m = (fabs.f64 ky)
ky\_s = (copysign.f64 #s(literal 1 binary64) ky)
(FPCore (ky_s kx ky_m th) :precision binary64 (* ky_s 0.0))

C

ky\_m = fabs(ky);
ky\_s = copysign(1.0, ky);
double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * 0.0;
}

Fortran

ky\_m = abs(ky)
ky\_s = copysign(1.0d0, ky)
real(8) function code(ky_s, kx, ky_m, th)
    real(8), intent (in) :: ky_s
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8), intent (in) :: th
    code = ky_s * 0.0d0
end function

Java

ky\_m = Math.abs(ky);
ky\_s = Math.copySign(1.0, ky);
public static double code(double ky_s, double kx, double ky_m, double th) {
	return ky_s * 0.0;
}

Python

ky\_m = math.fabs(ky)
ky\_s = math.copysign(1.0, ky)
def code(ky_s, kx, ky_m, th):
	return ky_s * 0.0

Julia

ky\_m = abs(ky)
ky\_s = copysign(1.0, ky)
function code(ky_s, kx, ky_m, th)
	return Float64(ky_s * 0.0)
end

MATLAB

ky\_m = abs(ky);
ky\_s = sign(ky) * abs(1.0);
function tmp = code(ky_s, kx, ky_m, th)
	tmp = ky_s * 0.0;
end

Wolfram

ky\_m = N[Abs[ky], $MachinePrecision]
ky\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[ky$95$s_, kx_, ky$95$m_, th_] := N[(ky$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. unpow2 91.5%

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

   2. sqr-neg 91.5%

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

   3. sin-neg 91.5%

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

   4. sin-neg 91.5%

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

   5. unpow2 91.5%

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

   6. associate-*l/ 90.2%

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

   7. associate-/l* 91.4%

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

   8. +-commutative 91.4%

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

   9. unpow2 91.4%

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

   10. sin-neg 91.4%

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

   11. sin-neg 91.4%

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

3. Simplified 99.6%

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

5. Taylor expanded in kx around 0 23.9%

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

   1. add-log-exp 20.9%

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

7. Applied egg-rr 20.9%

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

8. Taylor expanded in th around 0 11.3%

   \[\leadsto \log \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 11.3%

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 11.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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)))