inverse-phi

Percentage Accurate: 96.7% → 99.8%

Time: 6.2s

Alternatives: 21

Speedup: 0.9×

Specification

Math

\[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (atan
 (/ (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt))) (* one_es sa))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / (one_es * sa)));
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    code = atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / (one_es * sa)))
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return Math.atan((((Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt))) / (one_es * sa)));
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	return math.atan((((math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))) / (one_es * sa)))

Julia

function code(lamdp, lamt, ca, one_es, sa)
	return atan(Float64(Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt))) / Float64(one_es * sa)))
end

MATLAB

function tmp = code(lamdp, lamt, ca, one_es, sa)
	tmp = atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / (one_es * sa)));
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	atan(((((tan(lamdp)) * (cos(lamt))) - (ca * (sin(lamt)))) / (one_es * sa)))
END code

Initial Program: 96.7% accurate, 1.0× speedup

Math

\[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (atan
 (/ (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt))) (* one_es sa))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / (one_es * sa)));
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    code = atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / (one_es * sa)))
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return Math.atan((((Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt))) / (one_es * sa)));
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	return math.atan((((math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))) / (one_es * sa)))

Julia

function code(lamdp, lamt, ca, one_es, sa)
	return atan(Float64(Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt))) / Float64(one_es * sa)))
end

MATLAB

function tmp = code(lamdp, lamt, ca, one_es, sa)
	tmp = atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / (one_es * sa)));
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	atan(((((tan(lamdp)) * (cos(lamt))) - (ca * (sin(lamt)))) / (one_es * sa)))
END code

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \tan^{-1} \left(\frac{\sin lamt \cdot ca - \cos lamt \cdot \tan lamdp}{\mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)} \cdot \frac{-1}{\mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)}\right)\right) \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 one_es)
 (*
  (copysign 1.0 sa)
  (atan
   (*
    (/
     (- (* (sin lamt) ca) (* (cos lamt) (tan lamdp)))
     (fmin (fabs one_es) (fabs sa)))
    (/ -1.0 (fmax (fabs one_es) (fabs sa))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return copysign(1.0, one_es) * (copysign(1.0, sa) * atan(((((sin(lamt) * ca) - (cos(lamt) * tan(lamdp))) / fmin(fabs(one_es), fabs(sa))) * (-1.0 / fmax(fabs(one_es), fabs(sa))))));
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * Math.atan(((((Math.sin(lamt) * ca) - (Math.cos(lamt) * Math.tan(lamdp))) / fmin(Math.abs(one_es), Math.abs(sa))) * (-1.0 / fmax(Math.abs(one_es), Math.abs(sa))))));
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * math.atan(((((math.sin(lamt) * ca) - (math.cos(lamt) * math.tan(lamdp))) / fmin(math.fabs(one_es), math.fabs(sa))) * (-1.0 / fmax(math.fabs(one_es), math.fabs(sa))))))

Julia

function code(lamdp, lamt, ca, one_es, sa)
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * atan(Float64(Float64(Float64(Float64(sin(lamt) * ca) - Float64(cos(lamt) * tan(lamdp))) / fmin(abs(one_es), abs(sa))) * Float64(-1.0 / fmax(abs(one_es), abs(sa)))))))
end

MATLAB

function tmp = code(lamdp, lamt, ca, one_es, sa)
	tmp = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * atan(((((sin(lamt) * ca) - (cos(lamt) * tan(lamdp))) / min(abs(one_es), abs(sa))) * (-1.0 / max(abs(one_es), abs(sa))))));
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[ArcTan[N[(N[(N[(N[(N[Sin[lamt], $MachinePrecision] * ca), $MachinePrecision] - N[(N[Cos[lamt], $MachinePrecision] * N[Tan[lamdp], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

   \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{\color{blue}{one\_es \cdot sa}}\right) \]

   3. associate-/r* N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}}{sa}\right)} \]

   4. frac-2neg N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)}{\mathsf{neg}\left(sa\right)}\right)} \]

   5. mult-flip N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right)} \]

   7. distribute-neg-frac N/A

      \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)\right)}{one\_es}} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)\right)}{one\_es}} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   9. lift--.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)}\right)}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   10. sub-negate-rev N/A

       \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt - \tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   11. lower--.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt - \tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   13. *-commutative N/A

       \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\sin lamt \cdot ca} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\sin lamt \cdot ca} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   16. *-commutative N/A

       \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\cos lamt \cdot \tan lamdp}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\cos lamt \cdot \tan lamdp}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

3. Applied rewrites 97.9%

   \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\sin lamt \cdot ca - \cos lamt \cdot \tan lamdp}{one\_es} \cdot \frac{-1}{sa}\right)} \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp - \sin lamt \cdot ca}{\mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)}}{\mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)}\right)\right) \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (*
 (copysign 1.0 one_es)
 (*
  (copysign 1.0 sa)
  (atan
   (/
    (/
     (- (* (cos lamt) (tan lamdp)) (* (sin lamt) ca))
     (fmin (fabs one_es) (fabs sa)))
    (fmax (fabs one_es) (fabs sa)))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return copysign(1.0, one_es) * (copysign(1.0, sa) * atan(((((cos(lamt) * tan(lamdp)) - (sin(lamt) * ca)) / fmin(fabs(one_es), fabs(sa))) / fmax(fabs(one_es), fabs(sa)))));
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * Math.atan(((((Math.cos(lamt) * Math.tan(lamdp)) - (Math.sin(lamt) * ca)) / fmin(Math.abs(one_es), Math.abs(sa))) / fmax(Math.abs(one_es), Math.abs(sa)))));
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * math.atan(((((math.cos(lamt) * math.tan(lamdp)) - (math.sin(lamt) * ca)) / fmin(math.fabs(one_es), math.fabs(sa))) / fmax(math.fabs(one_es), math.fabs(sa)))))

Julia

function code(lamdp, lamt, ca, one_es, sa)
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * atan(Float64(Float64(Float64(Float64(cos(lamt) * tan(lamdp)) - Float64(sin(lamt) * ca)) / fmin(abs(one_es), abs(sa))) / fmax(abs(one_es), abs(sa))))))
end

MATLAB

function tmp = code(lamdp, lamt, ca, one_es, sa)
	tmp = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * atan(((((cos(lamt) * tan(lamdp)) - (sin(lamt) * ca)) / min(abs(one_es), abs(sa))) / max(abs(one_es), abs(sa)))));
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[ArcTan[N[(N[(N[(N[(N[Cos[lamt], $MachinePrecision] * N[Tan[lamdp], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lamt], $MachinePrecision] * ca), $MachinePrecision]), $MachinePrecision] / N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

   \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{\color{blue}{one\_es \cdot sa}}\right) \]

   3. associate-/r* N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}}{sa}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}}{sa}\right)} \]

   5. lower-/.f64 97.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}}}{sa}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{\color{blue}{\tan lamdp \cdot \cos lamt} - ca \cdot \sin lamt}{one\_es}}{sa}\right) \]

   7. *-commutative N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{\color{blue}{\cos lamt \cdot \tan lamdp} - ca \cdot \sin lamt}{one\_es}}{sa}\right) \]

   8. lower-*.f64 97.9%

      \[\leadsto \tan^{-1} \left(\frac{\frac{\color{blue}{\cos lamt \cdot \tan lamdp} - ca \cdot \sin lamt}{one\_es}}{sa}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp - \color{blue}{ca \cdot \sin lamt}}{one\_es}}{sa}\right) \]

   10. *-commutative N/A

       \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp - \color{blue}{\sin lamt \cdot ca}}{one\_es}}{sa}\right) \]

   11. lower-*.f64 97.9%

       \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp - \color{blue}{\sin lamt \cdot ca}}{one\_es}}{sa}\right) \]

3. Applied rewrites 97.9%

   \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\cos lamt \cdot \tan lamdp - \sin lamt \cdot ca}{one\_es}}{sa}\right)} \]
4. Add Preprocessing

Alternative 3: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \left(-ca\right) \cdot \sin lamt\\ t_2 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_3 := t\_2 \cdot t\_0\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\tan^{-1} \left(\frac{\mathsf{fma}\left(\tan lamdp, \cos lamt, t\_1\right)}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_1}{t\_0}}{t\_2}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmax (fabs one_es) (fabs sa)))
       (t_1 (* (- ca) (sin lamt)))
       (t_2 (fmin (fabs one_es) (fabs sa)))
       (t_3 (* t_2 t_0)))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_3 2e+269)
      (atan (/ (fma (tan lamdp) (cos lamt) t_1) t_3))
      (atan (/ (/ t_1 t_0) t_2)))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmax(fabs(one_es), fabs(sa));
	double t_1 = -ca * sin(lamt);
	double t_2 = fmin(fabs(one_es), fabs(sa));
	double t_3 = t_2 * t_0;
	double tmp;
	if (t_3 <= 2e+269) {
		tmp = atan((fma(tan(lamdp), cos(lamt), t_1) / t_3));
	} else {
		tmp = atan(((t_1 / t_0) / t_2));
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmax(abs(one_es), abs(sa))
	t_1 = Float64(Float64(-ca) * sin(lamt))
	t_2 = fmin(abs(one_es), abs(sa))
	t_3 = Float64(t_2 * t_0)
	tmp = 0.0
	if (t_3 <= 2e+269)
		tmp = atan(Float64(fma(tan(lamdp), cos(lamt), t_1) / t_3));
	else
		tmp = atan(Float64(Float64(t_1 / t_0) / t_2));
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 2e+269], N[ArcTan[N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision] + t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 one_es sa) < 2.0000000000000001e269

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}}{one\_es \cdot sa}\right) \]

      2. sub-flip N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\tan lamdp \cdot \cos lamt + \left(\mathsf{neg}\left(ca \cdot \sin lamt\right)\right)}}{one\_es \cdot sa}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\tan lamdp \cdot \cos lamt} + \left(\mathsf{neg}\left(ca \cdot \sin lamt\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\tan lamdp, \cos lamt, \mathsf{neg}\left(ca \cdot \sin lamt\right)\right)}}{one\_es \cdot sa}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{fma}\left(\tan lamdp, \cos lamt, \mathsf{neg}\left(\color{blue}{ca \cdot \sin lamt}\right)\right)}{one\_es \cdot sa}\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{fma}\left(\tan lamdp, \cos lamt, \color{blue}{\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin lamt}\right)}{one\_es \cdot sa}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{fma}\left(\tan lamdp, \cos lamt, \color{blue}{\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin lamt}\right)}{one\_es \cdot sa}\right) \]

      8. lower-neg.f64 96.7%

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{fma}\left(\tan lamdp, \cos lamt, \color{blue}{\left(-ca\right)} \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   3. Applied rewrites 96.7%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\tan lamdp, \cos lamt, \left(-ca\right) \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   if 2.0000000000000001e269 < (*.f64 one_es sa)

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_2 := t\_1 \cdot t\_0\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{t\_0}}{t\_1}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmax (fabs one_es) (fabs sa)))
       (t_1 (fmin (fabs one_es) (fabs sa)))
       (t_2 (* t_1 t_0)))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_2 2e+269)
      (atan (/ (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt))) t_2))
      (atan (/ (/ (* (- ca) (sin lamt)) t_0) t_1)))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmax(fabs(one_es), fabs(sa));
	double t_1 = fmin(fabs(one_es), fabs(sa));
	double t_2 = t_1 * t_0;
	double tmp;
	if (t_2 <= 2e+269) {
		tmp = atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / t_2));
	} else {
		tmp = atan((((-ca * sin(lamt)) / t_0) / t_1));
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_1 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_2 = t_1 * t_0;
	double tmp;
	if (t_2 <= 2e+269) {
		tmp = Math.atan((((Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt))) / t_2));
	} else {
		tmp = Math.atan((((-ca * Math.sin(lamt)) / t_0) / t_1));
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmax(math.fabs(one_es), math.fabs(sa))
	t_1 = fmin(math.fabs(one_es), math.fabs(sa))
	t_2 = t_1 * t_0
	tmp = 0
	if t_2 <= 2e+269:
		tmp = math.atan((((math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))) / t_2))
	else:
		tmp = math.atan((((-ca * math.sin(lamt)) / t_0) / t_1))
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmax(abs(one_es), abs(sa))
	t_1 = fmin(abs(one_es), abs(sa))
	t_2 = Float64(t_1 * t_0)
	tmp = 0.0
	if (t_2 <= 2e+269)
		tmp = atan(Float64(Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt))) / t_2));
	else
		tmp = atan(Float64(Float64(Float64(Float64(-ca) * sin(lamt)) / t_0) / t_1));
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = max(abs(one_es), abs(sa));
	t_1 = min(abs(one_es), abs(sa));
	t_2 = t_1 * t_0;
	tmp = 0.0;
	if (t_2 <= 2e+269)
		tmp = atan((((tan(lamdp) * cos(lamt)) - (ca * sin(lamt))) / t_2));
	else
		tmp = atan((((-ca * sin(lamt)) / t_0) / t_1));
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, 2e+269], N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 one_es sa) < 2.0000000000000001e269

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   if 2.0000000000000001e269 < (*.f64 one_es sa)

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := ca \cdot \sin lamt\\ t_2 := \tan lamdp \cdot \cos lamt - t\_1\\ t_3 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_4 := \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{t\_0}}{t\_3}\right)\\ t_5 := \tan^{-1} \left(\frac{\frac{1 \cdot \tan lamdp - \sin lamt \cdot ca}{t\_3}}{t\_0}\right)\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -50000:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq -0.09:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} \left(\frac{lamdp \cdot \cos lamt - t\_1}{t\_0 \cdot t\_3}\right)\\ \mathbf{elif}\;t\_2 \leq 0.48:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (* ca (sin lamt)))
       (t_2 (- (* (tan lamdp) (cos lamt)) t_1))
       (t_3 (fmax (fabs one_es) (fabs sa)))
       (t_4 (atan (/ (* (cos lamt) (/ (tan lamdp) t_0)) t_3)))
       (t_5
        (atan
         (/ (/ (- (* 1.0 (tan lamdp)) (* (sin lamt) ca)) t_3) t_0))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_2 -50000.0)
      t_5
      (if (<= t_2 -0.09)
        t_4
        (if (<= t_2 5e-10)
          (atan (/ (- (* lamdp (cos lamt)) t_1) (* t_0 t_3)))
          (if (<= t_2 0.48) t_4 t_5))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = ca * sin(lamt);
	double t_2 = (tan(lamdp) * cos(lamt)) - t_1;
	double t_3 = fmax(fabs(one_es), fabs(sa));
	double t_4 = atan(((cos(lamt) * (tan(lamdp) / t_0)) / t_3));
	double t_5 = atan(((((1.0 * tan(lamdp)) - (sin(lamt) * ca)) / t_3) / t_0));
	double tmp;
	if (t_2 <= -50000.0) {
		tmp = t_5;
	} else if (t_2 <= -0.09) {
		tmp = t_4;
	} else if (t_2 <= 5e-10) {
		tmp = atan((((lamdp * cos(lamt)) - t_1) / (t_0 * t_3)));
	} else if (t_2 <= 0.48) {
		tmp = t_4;
	} else {
		tmp = t_5;
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = ca * Math.sin(lamt);
	double t_2 = (Math.tan(lamdp) * Math.cos(lamt)) - t_1;
	double t_3 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_4 = Math.atan(((Math.cos(lamt) * (Math.tan(lamdp) / t_0)) / t_3));
	double t_5 = Math.atan(((((1.0 * Math.tan(lamdp)) - (Math.sin(lamt) * ca)) / t_3) / t_0));
	double tmp;
	if (t_2 <= -50000.0) {
		tmp = t_5;
	} else if (t_2 <= -0.09) {
		tmp = t_4;
	} else if (t_2 <= 5e-10) {
		tmp = Math.atan((((lamdp * Math.cos(lamt)) - t_1) / (t_0 * t_3)));
	} else if (t_2 <= 0.48) {
		tmp = t_4;
	} else {
		tmp = t_5;
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = ca * math.sin(lamt)
	t_2 = (math.tan(lamdp) * math.cos(lamt)) - t_1
	t_3 = fmax(math.fabs(one_es), math.fabs(sa))
	t_4 = math.atan(((math.cos(lamt) * (math.tan(lamdp) / t_0)) / t_3))
	t_5 = math.atan(((((1.0 * math.tan(lamdp)) - (math.sin(lamt) * ca)) / t_3) / t_0))
	tmp = 0
	if t_2 <= -50000.0:
		tmp = t_5
	elif t_2 <= -0.09:
		tmp = t_4
	elif t_2 <= 5e-10:
		tmp = math.atan((((lamdp * math.cos(lamt)) - t_1) / (t_0 * t_3)))
	elif t_2 <= 0.48:
		tmp = t_4
	else:
		tmp = t_5
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = Float64(ca * sin(lamt))
	t_2 = Float64(Float64(tan(lamdp) * cos(lamt)) - t_1)
	t_3 = fmax(abs(one_es), abs(sa))
	t_4 = atan(Float64(Float64(cos(lamt) * Float64(tan(lamdp) / t_0)) / t_3))
	t_5 = atan(Float64(Float64(Float64(Float64(1.0 * tan(lamdp)) - Float64(sin(lamt) * ca)) / t_3) / t_0))
	tmp = 0.0
	if (t_2 <= -50000.0)
		tmp = t_5;
	elseif (t_2 <= -0.09)
		tmp = t_4;
	elseif (t_2 <= 5e-10)
		tmp = atan(Float64(Float64(Float64(lamdp * cos(lamt)) - t_1) / Float64(t_0 * t_3)));
	elseif (t_2 <= 0.48)
		tmp = t_4;
	else
		tmp = t_5;
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = ca * sin(lamt);
	t_2 = (tan(lamdp) * cos(lamt)) - t_1;
	t_3 = max(abs(one_es), abs(sa));
	t_4 = atan(((cos(lamt) * (tan(lamdp) / t_0)) / t_3));
	t_5 = atan(((((1.0 * tan(lamdp)) - (sin(lamt) * ca)) / t_3) / t_0));
	tmp = 0.0;
	if (t_2 <= -50000.0)
		tmp = t_5;
	elseif (t_2 <= -0.09)
		tmp = t_4;
	elseif (t_2 <= 5e-10)
		tmp = atan((((lamdp * cos(lamt)) - t_1) / (t_0 * t_3)));
	elseif (t_2 <= 0.48)
		tmp = t_4;
	else
		tmp = t_5;
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[(N[Cos[lamt], $MachinePrecision] * N[(N[Tan[lamdp], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[N[(N[(N[(N[(1.0 * N[Tan[lamdp], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lamt], $MachinePrecision] * ca), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -50000.0], t$95$5, If[LessEqual[t$95$2, -0.09], t$95$4, If[LessEqual[t$95$2, 5e-10], N[ArcTan[N[(N[(N[(lamdp * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 0.48], t$95$4, t$95$5]]]]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4 or 0.47999999999999998 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 86.0%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\tan lamdp \cdot 1 - ca \cdot \sin lamt}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot \sin lamt}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot 1 - ca \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot 1 - ca \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      6. lower-/.f64 87.6%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\tan lamdp \cdot 1 - ca \cdot \sin lamt}{sa}}}{one\_es}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\color{blue}{\tan lamdp \cdot 1} - ca \cdot \sin lamt}{sa}}{one\_es}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\color{blue}{1 \cdot \tan lamdp} - ca \cdot \sin lamt}{sa}}{one\_es}\right) \]

      9. lower-*.f64 87.6%

         \[\leadsto \tan^{-1} \left(\frac{\frac{\color{blue}{1 \cdot \tan lamdp} - ca \cdot \sin lamt}{sa}}{one\_es}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{1 \cdot \tan lamdp - \color{blue}{ca \cdot \sin lamt}}{sa}}{one\_es}\right) \]

      11. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{1 \cdot \tan lamdp - \color{blue}{\sin lamt \cdot ca}}{sa}}{one\_es}\right) \]

      12. lift-*.f64 87.6%

          \[\leadsto \tan^{-1} \left(\frac{\frac{1 \cdot \tan lamdp - \color{blue}{\sin lamt \cdot ca}}{sa}}{one\_es}\right) \]

   6. Applied rewrites 87.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{1 \cdot \tan lamdp - \sin lamt \cdot ca}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -0.089999999999999997 or 5.0000000000000003e-10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 0.47999999999999998

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es} \cdot sa}\right) \]

      14. associate-/r* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{sa}\right) \]

      17. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

      19. lower-/.f64 62.7%

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

   if -0.089999999999999997 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000003e-10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{lamdp \cdot \cos lamt} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \color{blue}{\cos lamt} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

      2. lower-cos.f64 71.6%

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 71.6%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{lamdp \cdot \cos lamt} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \left(-ca\right) \cdot \sin lamt\\ t_2 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_3 := ca \cdot \sin lamt\\ t_4 := \tan lamdp \cdot \cos lamt - t\_3\\ t_5 := \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{t\_0}}{t\_2}\right)\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -50000:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_1}{t\_2}}{t\_0}\right)\\ \mathbf{elif}\;t\_4 \leq -0.09:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} \left(\frac{lamdp \cdot \cos lamt - t\_3}{t\_0 \cdot t\_2}\right)\\ \mathbf{elif}\;t\_4 \leq 10:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{t\_2}{\frac{t\_1}{t\_0}}}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (* (- ca) (sin lamt)))
       (t_2 (fmax (fabs one_es) (fabs sa)))
       (t_3 (* ca (sin lamt)))
       (t_4 (- (* (tan lamdp) (cos lamt)) t_3))
       (t_5 (atan (/ (* (cos lamt) (/ (tan lamdp) t_0)) t_2))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_4 -50000.0)
      (atan (/ (/ t_1 t_2) t_0))
      (if (<= t_4 -0.09)
        t_5
        (if (<= t_4 5e-10)
          (atan (/ (- (* lamdp (cos lamt)) t_3) (* t_0 t_2)))
          (if (<= t_4 10.0)
            t_5
            (atan (/ 1.0 (/ t_2 (/ t_1 t_0))))))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = -ca * sin(lamt);
	double t_2 = fmax(fabs(one_es), fabs(sa));
	double t_3 = ca * sin(lamt);
	double t_4 = (tan(lamdp) * cos(lamt)) - t_3;
	double t_5 = atan(((cos(lamt) * (tan(lamdp) / t_0)) / t_2));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = atan(((t_1 / t_2) / t_0));
	} else if (t_4 <= -0.09) {
		tmp = t_5;
	} else if (t_4 <= 5e-10) {
		tmp = atan((((lamdp * cos(lamt)) - t_3) / (t_0 * t_2)));
	} else if (t_4 <= 10.0) {
		tmp = t_5;
	} else {
		tmp = atan((1.0 / (t_2 / (t_1 / t_0))));
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = -ca * Math.sin(lamt);
	double t_2 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_3 = ca * Math.sin(lamt);
	double t_4 = (Math.tan(lamdp) * Math.cos(lamt)) - t_3;
	double t_5 = Math.atan(((Math.cos(lamt) * (Math.tan(lamdp) / t_0)) / t_2));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = Math.atan(((t_1 / t_2) / t_0));
	} else if (t_4 <= -0.09) {
		tmp = t_5;
	} else if (t_4 <= 5e-10) {
		tmp = Math.atan((((lamdp * Math.cos(lamt)) - t_3) / (t_0 * t_2)));
	} else if (t_4 <= 10.0) {
		tmp = t_5;
	} else {
		tmp = Math.atan((1.0 / (t_2 / (t_1 / t_0))));
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = -ca * math.sin(lamt)
	t_2 = fmax(math.fabs(one_es), math.fabs(sa))
	t_3 = ca * math.sin(lamt)
	t_4 = (math.tan(lamdp) * math.cos(lamt)) - t_3
	t_5 = math.atan(((math.cos(lamt) * (math.tan(lamdp) / t_0)) / t_2))
	tmp = 0
	if t_4 <= -50000.0:
		tmp = math.atan(((t_1 / t_2) / t_0))
	elif t_4 <= -0.09:
		tmp = t_5
	elif t_4 <= 5e-10:
		tmp = math.atan((((lamdp * math.cos(lamt)) - t_3) / (t_0 * t_2)))
	elif t_4 <= 10.0:
		tmp = t_5
	else:
		tmp = math.atan((1.0 / (t_2 / (t_1 / t_0))))
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = Float64(Float64(-ca) * sin(lamt))
	t_2 = fmax(abs(one_es), abs(sa))
	t_3 = Float64(ca * sin(lamt))
	t_4 = Float64(Float64(tan(lamdp) * cos(lamt)) - t_3)
	t_5 = atan(Float64(Float64(cos(lamt) * Float64(tan(lamdp) / t_0)) / t_2))
	tmp = 0.0
	if (t_4 <= -50000.0)
		tmp = atan(Float64(Float64(t_1 / t_2) / t_0));
	elseif (t_4 <= -0.09)
		tmp = t_5;
	elseif (t_4 <= 5e-10)
		tmp = atan(Float64(Float64(Float64(lamdp * cos(lamt)) - t_3) / Float64(t_0 * t_2)));
	elseif (t_4 <= 10.0)
		tmp = t_5;
	else
		tmp = atan(Float64(1.0 / Float64(t_2 / Float64(t_1 / t_0))));
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = -ca * sin(lamt);
	t_2 = max(abs(one_es), abs(sa));
	t_3 = ca * sin(lamt);
	t_4 = (tan(lamdp) * cos(lamt)) - t_3;
	t_5 = atan(((cos(lamt) * (tan(lamdp) / t_0)) / t_2));
	tmp = 0.0;
	if (t_4 <= -50000.0)
		tmp = atan(((t_1 / t_2) / t_0));
	elseif (t_4 <= -0.09)
		tmp = t_5;
	elseif (t_4 <= 5e-10)
		tmp = atan((((lamdp * cos(lamt)) - t_3) / (t_0 * t_2)));
	elseif (t_4 <= 10.0)
		tmp = t_5;
	else
		tmp = atan((1.0 / (t_2 / (t_1 / t_0))));
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[N[(N[(N[Cos[lamt], $MachinePrecision] * N[(N[Tan[lamdp], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -50000.0], N[ArcTan[N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, -0.09], t$95$5, If[LessEqual[t$95$4, 5e-10], N[ArcTan[N[(N[(N[(lamdp * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 10.0], t$95$5, N[ArcTan[N[(1.0 / N[(t$95$2 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -0.089999999999999997 or 5.0000000000000003e-10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es} \cdot sa}\right) \]

      14. associate-/r* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{sa}\right) \]

      17. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

      19. lower-/.f64 62.7%

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

   if -0.089999999999999997 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000003e-10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{lamdp \cdot \cos lamt} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \color{blue}{\cos lamt} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

      2. lower-cos.f64 71.6%

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 71.6%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{lamdp \cdot \cos lamt} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   if 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. div-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      5. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      7. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      9. lower-/.f64 63.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\frac{sa}{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \left(-ca\right) \cdot \sin lamt\\ t_2 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_3 := \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{t\_0}}{t\_2}\right)\\ t_4 := \tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -50000:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_1}{t\_2}}{t\_0}\right)\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-156}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot lamt}{t\_0 \cdot t\_2}\right)\\ \mathbf{elif}\;t\_4 \leq 10:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{t\_2}{\frac{t\_1}{t\_0}}}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (* (- ca) (sin lamt)))
       (t_2 (fmax (fabs one_es) (fabs sa)))
       (t_3 (atan (/ (* (cos lamt) (/ (tan lamdp) t_0)) t_2)))
       (t_4 (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt)))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_4 -50000.0)
      (atan (/ (/ t_1 t_2) t_0))
      (if (<= t_4 5e-156)
        t_3
        (if (<= t_4 5e-10)
          (atan (/ (- (* (tan lamdp) 1.0) (* ca lamt)) (* t_0 t_2)))
          (if (<= t_4 10.0)
            t_3
            (atan (/ 1.0 (/ t_2 (/ t_1 t_0))))))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = -ca * sin(lamt);
	double t_2 = fmax(fabs(one_es), fabs(sa));
	double t_3 = atan(((cos(lamt) * (tan(lamdp) / t_0)) / t_2));
	double t_4 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = atan(((t_1 / t_2) / t_0));
	} else if (t_4 <= 5e-156) {
		tmp = t_3;
	} else if (t_4 <= 5e-10) {
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (t_0 * t_2)));
	} else if (t_4 <= 10.0) {
		tmp = t_3;
	} else {
		tmp = atan((1.0 / (t_2 / (t_1 / t_0))));
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = -ca * Math.sin(lamt);
	double t_2 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_3 = Math.atan(((Math.cos(lamt) * (Math.tan(lamdp) / t_0)) / t_2));
	double t_4 = (Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = Math.atan(((t_1 / t_2) / t_0));
	} else if (t_4 <= 5e-156) {
		tmp = t_3;
	} else if (t_4 <= 5e-10) {
		tmp = Math.atan((((Math.tan(lamdp) * 1.0) - (ca * lamt)) / (t_0 * t_2)));
	} else if (t_4 <= 10.0) {
		tmp = t_3;
	} else {
		tmp = Math.atan((1.0 / (t_2 / (t_1 / t_0))));
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = -ca * math.sin(lamt)
	t_2 = fmax(math.fabs(one_es), math.fabs(sa))
	t_3 = math.atan(((math.cos(lamt) * (math.tan(lamdp) / t_0)) / t_2))
	t_4 = (math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))
	tmp = 0
	if t_4 <= -50000.0:
		tmp = math.atan(((t_1 / t_2) / t_0))
	elif t_4 <= 5e-156:
		tmp = t_3
	elif t_4 <= 5e-10:
		tmp = math.atan((((math.tan(lamdp) * 1.0) - (ca * lamt)) / (t_0 * t_2)))
	elif t_4 <= 10.0:
		tmp = t_3
	else:
		tmp = math.atan((1.0 / (t_2 / (t_1 / t_0))))
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = Float64(Float64(-ca) * sin(lamt))
	t_2 = fmax(abs(one_es), abs(sa))
	t_3 = atan(Float64(Float64(cos(lamt) * Float64(tan(lamdp) / t_0)) / t_2))
	t_4 = Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt)))
	tmp = 0.0
	if (t_4 <= -50000.0)
		tmp = atan(Float64(Float64(t_1 / t_2) / t_0));
	elseif (t_4 <= 5e-156)
		tmp = t_3;
	elseif (t_4 <= 5e-10)
		tmp = atan(Float64(Float64(Float64(tan(lamdp) * 1.0) - Float64(ca * lamt)) / Float64(t_0 * t_2)));
	elseif (t_4 <= 10.0)
		tmp = t_3;
	else
		tmp = atan(Float64(1.0 / Float64(t_2 / Float64(t_1 / t_0))));
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = -ca * sin(lamt);
	t_2 = max(abs(one_es), abs(sa));
	t_3 = atan(((cos(lamt) * (tan(lamdp) / t_0)) / t_2));
	t_4 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	tmp = 0.0;
	if (t_4 <= -50000.0)
		tmp = atan(((t_1 / t_2) / t_0));
	elseif (t_4 <= 5e-156)
		tmp = t_3;
	elseif (t_4 <= 5e-10)
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (t_0 * t_2)));
	elseif (t_4 <= 10.0)
		tmp = t_3;
	else
		tmp = atan((1.0 / (t_2 / (t_1 / t_0))));
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(N[Cos[lamt], $MachinePrecision] * N[(N[Tan[lamdp], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -50000.0], N[ArcTan[N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e-156], t$95$3, If[LessEqual[t$95$4, 5e-10], N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * 1.0), $MachinePrecision] - N[(ca * lamt), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 10.0], t$95$3, N[ArcTan[N[(1.0 / N[(t$95$2 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000001e-156 or 5.0000000000000003e-10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es} \cdot sa}\right) \]

      14. associate-/r* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\cos lamt \cdot \tan lamdp}{one\_es}}{sa}\right) \]

      17. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

      19. lower-/.f64 62.7%

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{sa}\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \frac{\tan lamdp}{one\_es}}{\color{blue}{sa}}\right) \]

   if 5.0000000000000001e-156 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000003e-10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 86.0%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 67.2%

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot \color{blue}{lamt}}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 67.2%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]

   if 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. div-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      5. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      7. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      9. lower-/.f64 63.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\frac{sa}{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(-ca\right) \cdot \sin lamt\\ t_1 := \tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{sa}}{one\_es}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot one\_es}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot lamt}{one\_es \cdot sa}\right)\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;\tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{sa \cdot one\_es}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{sa}{\frac{t\_0}{one\_es}}}\right)\\ \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (- ca) (sin lamt)))
       (t_1 (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt)))))
  (if (<= t_1 -50000.0)
    (atan (/ (/ t_0 sa) one_es))
    (if (<= t_1 5e-156)
      (atan (* (cos lamt) (/ (tan lamdp) (* sa one_es))))
      (if (<= t_1 5e-10)
        (atan (/ (- (* (tan lamdp) 1.0) (* ca lamt)) (* one_es sa)))
        (if (<= t_1 10.0)
          (atan (/ (* (cos lamt) (tan lamdp)) (* sa one_es)))
          (atan (/ 1.0 (/ sa (/ t_0 one_es))))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * sin(lamt);
	double t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = atan(((t_0 / sa) / one_es));
	} else if (t_1 <= 5e-156) {
		tmp = atan((cos(lamt) * (tan(lamdp) / (sa * one_es))));
	} else if (t_1 <= 5e-10) {
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	} else if (t_1 <= 10.0) {
		tmp = atan(((cos(lamt) * tan(lamdp)) / (sa * one_es)));
	} else {
		tmp = atan((1.0 / (sa / (t_0 / one_es))));
	}
	return tmp;
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -ca * sin(lamt)
    t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt))
    if (t_1 <= (-50000.0d0)) then
        tmp = atan(((t_0 / sa) / one_es))
    else if (t_1 <= 5d-156) then
        tmp = atan((cos(lamt) * (tan(lamdp) / (sa * one_es))))
    else if (t_1 <= 5d-10) then
        tmp = atan((((tan(lamdp) * 1.0d0) - (ca * lamt)) / (one_es * sa)))
    else if (t_1 <= 10.0d0) then
        tmp = atan(((cos(lamt) * tan(lamdp)) / (sa * one_es)))
    else
        tmp = atan((1.0d0 / (sa / (t_0 / one_es))))
    end if
    code = tmp
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * Math.sin(lamt);
	double t_1 = (Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt));
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = Math.atan(((t_0 / sa) / one_es));
	} else if (t_1 <= 5e-156) {
		tmp = Math.atan((Math.cos(lamt) * (Math.tan(lamdp) / (sa * one_es))));
	} else if (t_1 <= 5e-10) {
		tmp = Math.atan((((Math.tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	} else if (t_1 <= 10.0) {
		tmp = Math.atan(((Math.cos(lamt) * Math.tan(lamdp)) / (sa * one_es)));
	} else {
		tmp = Math.atan((1.0 / (sa / (t_0 / one_es))));
	}
	return tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = -ca * math.sin(lamt)
	t_1 = (math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))
	tmp = 0
	if t_1 <= -50000.0:
		tmp = math.atan(((t_0 / sa) / one_es))
	elif t_1 <= 5e-156:
		tmp = math.atan((math.cos(lamt) * (math.tan(lamdp) / (sa * one_es))))
	elif t_1 <= 5e-10:
		tmp = math.atan((((math.tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)))
	elif t_1 <= 10.0:
		tmp = math.atan(((math.cos(lamt) * math.tan(lamdp)) / (sa * one_es)))
	else:
		tmp = math.atan((1.0 / (sa / (t_0 / one_es))))
	return tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = Float64(Float64(-ca) * sin(lamt))
	t_1 = Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt)))
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = atan(Float64(Float64(t_0 / sa) / one_es));
	elseif (t_1 <= 5e-156)
		tmp = atan(Float64(cos(lamt) * Float64(tan(lamdp) / Float64(sa * one_es))));
	elseif (t_1 <= 5e-10)
		tmp = atan(Float64(Float64(Float64(tan(lamdp) * 1.0) - Float64(ca * lamt)) / Float64(one_es * sa)));
	elseif (t_1 <= 10.0)
		tmp = atan(Float64(Float64(cos(lamt) * tan(lamdp)) / Float64(sa * one_es)));
	else
		tmp = atan(Float64(1.0 / Float64(sa / Float64(t_0 / one_es))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = -ca * sin(lamt);
	t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	tmp = 0.0;
	if (t_1 <= -50000.0)
		tmp = atan(((t_0 / sa) / one_es));
	elseif (t_1 <= 5e-156)
		tmp = atan((cos(lamt) * (tan(lamdp) / (sa * one_es))));
	elseif (t_1 <= 5e-10)
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	elseif (t_1 <= 10.0)
		tmp = atan(((cos(lamt) * tan(lamdp)) / (sa * one_es)));
	else
		tmp = atan((1.0 / (sa / (t_0 / one_es))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000.0], N[ArcTan[N[(N[(t$95$0 / sa), $MachinePrecision] / one$95$es), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e-156], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(N[Tan[lamdp], $MachinePrecision] / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e-10], N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * 1.0), $MachinePrecision] - N[(ca * lamt), $MachinePrecision]), $MachinePrecision] / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 10.0], N[ArcTan[N[(N[(N[Cos[lamt], $MachinePrecision] * N[Tan[lamdp], $MachinePrecision]), $MachinePrecision] / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 / N[(sa / N[(t$95$0 / one$95$es), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	LET t_0 = ((- ca) * (sin(lamt))) IN
		LET t_1 = (((tan(lamdp)) * (cos(lamt))) - (ca * (sin(lamt)))) IN
			LET tmp_3 = IF (t_1 <= (10)) THEN (atan((((cos(lamt)) * (tan(lamdp))) / (sa * one_es)))) ELSE (atan(((1) / (sa / (t_0 / one_es))))) ENDIF IN
			LET tmp_2 = IF (t_1 <= (50000000000000003114079572888992820944853434639298939146101474761962890625e-83)) THEN (atan(((((tan(lamdp)) * (1)) - (ca * lamt)) / (one_es * sa)))) ELSE tmp_3 ENDIF IN
			LET tmp_1 = IF (t_1 <= (50000000000000000715540317304108006029470210492243123761983918126875935262850296510107706907452510377550525720562888354785121499372493310097280340714982147141129125603356728531036914851254511846031568594152234841426192195868945903408443151863112545281109972151090996482250140895178763654401487741162529419075433079988589340884859636228166246584438021605672655567875796727046744383216037022066302597522735595703125e-568)) THEN (atan(((cos(lamt)) * ((tan(lamdp)) / (sa * one_es))))) ELSE tmp_2 ENDIF IN
			LET tmp = IF (t_1 <= (-5e4)) THEN (atan(((t_0 / sa) / one_es))) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 5 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000001e-156

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   if 5.0000000000000001e-156 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000003e-10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 86.0%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 67.2%

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot \color{blue}{lamt}}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 67.2%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]

   if 5.0000000000000003e-10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{sa \cdot one\_es}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{sa \cdot one\_es}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{sa} \cdot one\_es}\right) \]

      5. lower-/.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{sa \cdot one\_es}}\right) \]

   8. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{sa \cdot one\_es}}\right) \]

   if 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. div-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      5. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      7. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      9. lower-/.f64 63.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\frac{sa}{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(-ca\right) \cdot \sin lamt\\ t_1 := \tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\\ t_2 := \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot one\_es}\right)\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{sa}}{one\_es}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot lamt}{one\_es \cdot sa}\right)\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{sa}{\frac{t\_0}{one\_es}}}\right)\\ \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (- ca) (sin lamt)))
       (t_1 (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt))))
       (t_2 (atan (* (cos lamt) (/ (tan lamdp) (* sa one_es))))))
  (if (<= t_1 -50000.0)
    (atan (/ (/ t_0 sa) one_es))
    (if (<= t_1 5e-156)
      t_2
      (if (<= t_1 5e-10)
        (atan (/ (- (* (tan lamdp) 1.0) (* ca lamt)) (* one_es sa)))
        (if (<= t_1 10.0)
          t_2
          (atan (/ 1.0 (/ sa (/ t_0 one_es))))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * sin(lamt);
	double t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	double t_2 = atan((cos(lamt) * (tan(lamdp) / (sa * one_es))));
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = atan(((t_0 / sa) / one_es));
	} else if (t_1 <= 5e-156) {
		tmp = t_2;
	} else if (t_1 <= 5e-10) {
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	} else if (t_1 <= 10.0) {
		tmp = t_2;
	} else {
		tmp = atan((1.0 / (sa / (t_0 / one_es))));
	}
	return tmp;
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = -ca * sin(lamt)
    t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt))
    t_2 = atan((cos(lamt) * (tan(lamdp) / (sa * one_es))))
    if (t_1 <= (-50000.0d0)) then
        tmp = atan(((t_0 / sa) / one_es))
    else if (t_1 <= 5d-156) then
        tmp = t_2
    else if (t_1 <= 5d-10) then
        tmp = atan((((tan(lamdp) * 1.0d0) - (ca * lamt)) / (one_es * sa)))
    else if (t_1 <= 10.0d0) then
        tmp = t_2
    else
        tmp = atan((1.0d0 / (sa / (t_0 / one_es))))
    end if
    code = tmp
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * Math.sin(lamt);
	double t_1 = (Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt));
	double t_2 = Math.atan((Math.cos(lamt) * (Math.tan(lamdp) / (sa * one_es))));
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = Math.atan(((t_0 / sa) / one_es));
	} else if (t_1 <= 5e-156) {
		tmp = t_2;
	} else if (t_1 <= 5e-10) {
		tmp = Math.atan((((Math.tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	} else if (t_1 <= 10.0) {
		tmp = t_2;
	} else {
		tmp = Math.atan((1.0 / (sa / (t_0 / one_es))));
	}
	return tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = -ca * math.sin(lamt)
	t_1 = (math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))
	t_2 = math.atan((math.cos(lamt) * (math.tan(lamdp) / (sa * one_es))))
	tmp = 0
	if t_1 <= -50000.0:
		tmp = math.atan(((t_0 / sa) / one_es))
	elif t_1 <= 5e-156:
		tmp = t_2
	elif t_1 <= 5e-10:
		tmp = math.atan((((math.tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)))
	elif t_1 <= 10.0:
		tmp = t_2
	else:
		tmp = math.atan((1.0 / (sa / (t_0 / one_es))))
	return tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = Float64(Float64(-ca) * sin(lamt))
	t_1 = Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt)))
	t_2 = atan(Float64(cos(lamt) * Float64(tan(lamdp) / Float64(sa * one_es))))
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = atan(Float64(Float64(t_0 / sa) / one_es));
	elseif (t_1 <= 5e-156)
		tmp = t_2;
	elseif (t_1 <= 5e-10)
		tmp = atan(Float64(Float64(Float64(tan(lamdp) * 1.0) - Float64(ca * lamt)) / Float64(one_es * sa)));
	elseif (t_1 <= 10.0)
		tmp = t_2;
	else
		tmp = atan(Float64(1.0 / Float64(sa / Float64(t_0 / one_es))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = -ca * sin(lamt);
	t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	t_2 = atan((cos(lamt) * (tan(lamdp) / (sa * one_es))));
	tmp = 0.0;
	if (t_1 <= -50000.0)
		tmp = atan(((t_0 / sa) / one_es));
	elseif (t_1 <= 5e-156)
		tmp = t_2;
	elseif (t_1 <= 5e-10)
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	elseif (t_1 <= 10.0)
		tmp = t_2;
	else
		tmp = atan((1.0 / (sa / (t_0 / one_es))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(N[Tan[lamdp], $MachinePrecision] / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -50000.0], N[ArcTan[N[(N[(t$95$0 / sa), $MachinePrecision] / one$95$es), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e-156], t$95$2, If[LessEqual[t$95$1, 5e-10], N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * 1.0), $MachinePrecision] - N[(ca * lamt), $MachinePrecision]), $MachinePrecision] / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 10.0], t$95$2, N[ArcTan[N[(1.0 / N[(sa / N[(t$95$0 / one$95$es), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	LET t_0 = ((- ca) * (sin(lamt))) IN
		LET t_1 = (((tan(lamdp)) * (cos(lamt))) - (ca * (sin(lamt)))) IN
			LET t_2 = (atan(((cos(lamt)) * ((tan(lamdp)) / (sa * one_es))))) IN
				LET tmp_3 = IF (t_1 <= (10)) THEN t_2 ELSE (atan(((1) / (sa / (t_0 / one_es))))) ENDIF IN
				LET tmp_2 = IF (t_1 <= (50000000000000003114079572888992820944853434639298939146101474761962890625e-83)) THEN (atan(((((tan(lamdp)) * (1)) - (ca * lamt)) / (one_es * sa)))) ELSE tmp_3 ENDIF IN
				LET tmp_1 = IF (t_1 <= (50000000000000000715540317304108006029470210492243123761983918126875935262850296510107706907452510377550525720562888354785121499372493310097280340714982147141129125603356728531036914851254511846031568594152234841426192195868945903408443151863112545281109972151090996482250140895178763654401487741162529419075433079988589340884859636228166246584438021605672655567875796727046744383216037022066302597522735595703125e-568)) THEN t_2 ELSE tmp_2 ENDIF IN
				LET tmp = IF (t_1 <= (-5e4)) THEN (atan(((t_0 / sa) / one_es))) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000001e-156 or 5.0000000000000003e-10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   if 5.0000000000000001e-156 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 5.0000000000000003e-10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 86.0%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 67.2%

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot \color{blue}{lamt}}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 67.2%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]

   if 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. div-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      5. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      7. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      9. lower-/.f64 63.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\frac{sa}{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(-ca\right) \cdot \sin lamt\\ t_1 := \tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\\ \mathbf{if}\;t\_1 \leq -2.22 \cdot 10^{+21}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{sa}}{one\_es}\right)\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot lamt}{one\_es \cdot sa}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{sa}{\frac{t\_0}{one\_es}}}\right)\\ \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (- ca) (sin lamt)))
       (t_1 (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt)))))
  (if (<= t_1 -2.22e+21)
    (atan (/ (/ t_0 sa) one_es))
    (if (<= t_1 200.0)
      (atan (/ (- (* (tan lamdp) 1.0) (* ca lamt)) (* one_es sa)))
      (atan (/ 1.0 (/ sa (/ t_0 one_es))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * sin(lamt);
	double t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	double tmp;
	if (t_1 <= -2.22e+21) {
		tmp = atan(((t_0 / sa) / one_es));
	} else if (t_1 <= 200.0) {
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	} else {
		tmp = atan((1.0 / (sa / (t_0 / one_es))));
	}
	return tmp;
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -ca * sin(lamt)
    t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt))
    if (t_1 <= (-2.22d+21)) then
        tmp = atan(((t_0 / sa) / one_es))
    else if (t_1 <= 200.0d0) then
        tmp = atan((((tan(lamdp) * 1.0d0) - (ca * lamt)) / (one_es * sa)))
    else
        tmp = atan((1.0d0 / (sa / (t_0 / one_es))))
    end if
    code = tmp
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * Math.sin(lamt);
	double t_1 = (Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt));
	double tmp;
	if (t_1 <= -2.22e+21) {
		tmp = Math.atan(((t_0 / sa) / one_es));
	} else if (t_1 <= 200.0) {
		tmp = Math.atan((((Math.tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	} else {
		tmp = Math.atan((1.0 / (sa / (t_0 / one_es))));
	}
	return tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = -ca * math.sin(lamt)
	t_1 = (math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))
	tmp = 0
	if t_1 <= -2.22e+21:
		tmp = math.atan(((t_0 / sa) / one_es))
	elif t_1 <= 200.0:
		tmp = math.atan((((math.tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)))
	else:
		tmp = math.atan((1.0 / (sa / (t_0 / one_es))))
	return tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = Float64(Float64(-ca) * sin(lamt))
	t_1 = Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt)))
	tmp = 0.0
	if (t_1 <= -2.22e+21)
		tmp = atan(Float64(Float64(t_0 / sa) / one_es));
	elseif (t_1 <= 200.0)
		tmp = atan(Float64(Float64(Float64(tan(lamdp) * 1.0) - Float64(ca * lamt)) / Float64(one_es * sa)));
	else
		tmp = atan(Float64(1.0 / Float64(sa / Float64(t_0 / one_es))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = -ca * sin(lamt);
	t_1 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	tmp = 0.0;
	if (t_1 <= -2.22e+21)
		tmp = atan(((t_0 / sa) / one_es));
	elseif (t_1 <= 200.0)
		tmp = atan((((tan(lamdp) * 1.0) - (ca * lamt)) / (one_es * sa)));
	else
		tmp = atan((1.0 / (sa / (t_0 / one_es))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2.22e+21], N[ArcTan[N[(N[(t$95$0 / sa), $MachinePrecision] / one$95$es), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[ArcTan[N[(N[(N[(N[Tan[lamdp], $MachinePrecision] * 1.0), $MachinePrecision] - N[(ca * lamt), $MachinePrecision]), $MachinePrecision] / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 / N[(sa / N[(t$95$0 / one$95$es), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	LET t_0 = ((- ca) * (sin(lamt))) IN
		LET t_1 = (((tan(lamdp)) * (cos(lamt))) - (ca * (sin(lamt)))) IN
			LET tmp_1 = IF (t_1 <= (200)) THEN (atan(((((tan(lamdp)) * (1)) - (ca * lamt)) / (one_es * sa)))) ELSE (atan(((1) / (sa / (t_0 / one_es))))) ENDIF IN
			LET tmp = IF (t_1 <= (-222e19)) THEN (atan(((t_0 / sa) / one_es))) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -2.22e21

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -2.22e21 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 200

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 86.0%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \color{blue}{1} - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 67.2%

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - ca \cdot \color{blue}{lamt}}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 67.2%

      \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot 1 - \color{blue}{ca \cdot lamt}}{one\_es \cdot sa}\right) \]

   if 200 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. div-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      5. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      7. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      9. lower-/.f64 63.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\frac{sa}{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \left(-ca\right) \cdot \sin lamt\\ t_2 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_3 := ca \cdot \sin lamt\\ t_4 := \tan lamdp \cdot \cos lamt - t\_3\\ t_5 := \tan^{-1} \left(1 \cdot \frac{\tan lamdp}{t\_2 \cdot t\_0}\right)\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -50000:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_1}{t\_2}}{t\_0}\right)\\ \mathbf{elif}\;t\_4 \leq -0.09:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 10^{-172}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{t\_0 \cdot t\_2}\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\tan^{-1} \left(\frac{t\_3}{t\_0} \cdot \frac{-1}{t\_2}\right)\\ \mathbf{elif}\;t\_4 \leq 10:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{t\_2}{\frac{t\_1}{t\_0}}}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (* (- ca) (sin lamt)))
       (t_2 (fmax (fabs one_es) (fabs sa)))
       (t_3 (* ca (sin lamt)))
       (t_4 (- (* (tan lamdp) (cos lamt)) t_3))
       (t_5 (atan (* 1.0 (/ (tan lamdp) (* t_2 t_0))))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_4 -50000.0)
      (atan (/ (/ t_1 t_2) t_0))
      (if (<= t_4 -0.09)
        t_5
        (if (<= t_4 1e-172)
          (atan (* (cos lamt) (/ lamdp (* t_0 t_2))))
          (if (<= t_4 2e-42)
            (atan (* (/ t_3 t_0) (/ -1.0 t_2)))
            (if (<= t_4 10.0)
              t_5
              (atan (/ 1.0 (/ t_2 (/ t_1 t_0)))))))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = -ca * sin(lamt);
	double t_2 = fmax(fabs(one_es), fabs(sa));
	double t_3 = ca * sin(lamt);
	double t_4 = (tan(lamdp) * cos(lamt)) - t_3;
	double t_5 = atan((1.0 * (tan(lamdp) / (t_2 * t_0))));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = atan(((t_1 / t_2) / t_0));
	} else if (t_4 <= -0.09) {
		tmp = t_5;
	} else if (t_4 <= 1e-172) {
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_2))));
	} else if (t_4 <= 2e-42) {
		tmp = atan(((t_3 / t_0) * (-1.0 / t_2)));
	} else if (t_4 <= 10.0) {
		tmp = t_5;
	} else {
		tmp = atan((1.0 / (t_2 / (t_1 / t_0))));
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = -ca * Math.sin(lamt);
	double t_2 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_3 = ca * Math.sin(lamt);
	double t_4 = (Math.tan(lamdp) * Math.cos(lamt)) - t_3;
	double t_5 = Math.atan((1.0 * (Math.tan(lamdp) / (t_2 * t_0))));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = Math.atan(((t_1 / t_2) / t_0));
	} else if (t_4 <= -0.09) {
		tmp = t_5;
	} else if (t_4 <= 1e-172) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (t_0 * t_2))));
	} else if (t_4 <= 2e-42) {
		tmp = Math.atan(((t_3 / t_0) * (-1.0 / t_2)));
	} else if (t_4 <= 10.0) {
		tmp = t_5;
	} else {
		tmp = Math.atan((1.0 / (t_2 / (t_1 / t_0))));
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = -ca * math.sin(lamt)
	t_2 = fmax(math.fabs(one_es), math.fabs(sa))
	t_3 = ca * math.sin(lamt)
	t_4 = (math.tan(lamdp) * math.cos(lamt)) - t_3
	t_5 = math.atan((1.0 * (math.tan(lamdp) / (t_2 * t_0))))
	tmp = 0
	if t_4 <= -50000.0:
		tmp = math.atan(((t_1 / t_2) / t_0))
	elif t_4 <= -0.09:
		tmp = t_5
	elif t_4 <= 1e-172:
		tmp = math.atan((math.cos(lamt) * (lamdp / (t_0 * t_2))))
	elif t_4 <= 2e-42:
		tmp = math.atan(((t_3 / t_0) * (-1.0 / t_2)))
	elif t_4 <= 10.0:
		tmp = t_5
	else:
		tmp = math.atan((1.0 / (t_2 / (t_1 / t_0))))
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = Float64(Float64(-ca) * sin(lamt))
	t_2 = fmax(abs(one_es), abs(sa))
	t_3 = Float64(ca * sin(lamt))
	t_4 = Float64(Float64(tan(lamdp) * cos(lamt)) - t_3)
	t_5 = atan(Float64(1.0 * Float64(tan(lamdp) / Float64(t_2 * t_0))))
	tmp = 0.0
	if (t_4 <= -50000.0)
		tmp = atan(Float64(Float64(t_1 / t_2) / t_0));
	elseif (t_4 <= -0.09)
		tmp = t_5;
	elseif (t_4 <= 1e-172)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(t_0 * t_2))));
	elseif (t_4 <= 2e-42)
		tmp = atan(Float64(Float64(t_3 / t_0) * Float64(-1.0 / t_2)));
	elseif (t_4 <= 10.0)
		tmp = t_5;
	else
		tmp = atan(Float64(1.0 / Float64(t_2 / Float64(t_1 / t_0))));
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = -ca * sin(lamt);
	t_2 = max(abs(one_es), abs(sa));
	t_3 = ca * sin(lamt);
	t_4 = (tan(lamdp) * cos(lamt)) - t_3;
	t_5 = atan((1.0 * (tan(lamdp) / (t_2 * t_0))));
	tmp = 0.0;
	if (t_4 <= -50000.0)
		tmp = atan(((t_1 / t_2) / t_0));
	elseif (t_4 <= -0.09)
		tmp = t_5;
	elseif (t_4 <= 1e-172)
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_2))));
	elseif (t_4 <= 2e-42)
		tmp = atan(((t_3 / t_0) * (-1.0 / t_2)));
	elseif (t_4 <= 10.0)
		tmp = t_5;
	else
		tmp = atan((1.0 / (t_2 / (t_1 / t_0))));
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[N[(1.0 * N[(N[Tan[lamdp], $MachinePrecision] / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -50000.0], N[ArcTan[N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, -0.09], t$95$5, If[LessEqual[t$95$4, 1e-172], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e-42], N[ArcTan[N[(N[(t$95$3 / t$95$0), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 10.0], t$95$5, N[ArcTan[N[(1.0 / N[(t$95$2 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -0.089999999999999997 or 2.0000000000000001e-42 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(1 \cdot \frac{\color{blue}{\tan lamdp}}{sa \cdot one\_es}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 53.3%

      \[\leadsto \tan^{-1} \left(1 \cdot \frac{\color{blue}{\tan lamdp}}{sa \cdot one\_es}\right) \]

   if -0.089999999999999997 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 1e-172

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

   if 1e-172 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 2.0000000000000001e-42

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}}{sa}\right)} \]

      4. frac-2neg N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)}{\mathsf{neg}\left(sa\right)}\right)} \]

      5. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right)} \]

      7. distribute-neg-frac N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)\right)}{one\_es}} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)\right)}{one\_es}} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      9. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)}\right)}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt - \tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt - \tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\sin lamt \cdot ca} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\sin lamt \cdot ca} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\cos lamt \cdot \tan lamdp}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\cos lamt \cdot \tan lamdp}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   3. Applied rewrites 97.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\sin lamt \cdot ca - \cos lamt \cdot \tan lamdp}{one\_es} \cdot \frac{-1}{sa}\right)} \]

   4. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt}}{one\_es} \cdot \frac{-1}{sa}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{ca \cdot \color{blue}{\sin lamt}}{one\_es} \cdot \frac{-1}{sa}\right) \]

      2. lower-sin.f64 64.1%

         \[\leadsto \tan^{-1} \left(\frac{ca \cdot \sin lamt}{one\_es} \cdot \frac{-1}{sa}\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt}}{one\_es} \cdot \frac{-1}{sa}\right) \]

   if 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. div-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      5. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}\right)} \]

      7. remove-sound-/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\frac{sa}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

      9. lower-/.f64 63.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\frac{sa}{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}}\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{sa}{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_2 := \tan^{-1} \left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{t\_1}}{t\_0}\right)\\ t_3 := ca \cdot \sin lamt\\ t_4 := \tan lamdp \cdot \cos lamt - t\_3\\ t_5 := \tan^{-1} \left(1 \cdot \frac{\tan lamdp}{t\_1 \cdot t\_0}\right)\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -50000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq -0.09:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 10^{-172}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{t\_0 \cdot t\_1}\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\tan^{-1} \left(\frac{t\_3}{t\_0} \cdot \frac{-1}{t\_1}\right)\\ \mathbf{elif}\;t\_4 \leq 10:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (fmax (fabs one_es) (fabs sa)))
       (t_2 (atan (/ (/ (* (- ca) (sin lamt)) t_1) t_0)))
       (t_3 (* ca (sin lamt)))
       (t_4 (- (* (tan lamdp) (cos lamt)) t_3))
       (t_5 (atan (* 1.0 (/ (tan lamdp) (* t_1 t_0))))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_4 -50000.0)
      t_2
      (if (<= t_4 -0.09)
        t_5
        (if (<= t_4 1e-172)
          (atan (* (cos lamt) (/ lamdp (* t_0 t_1))))
          (if (<= t_4 2e-42)
            (atan (* (/ t_3 t_0) (/ -1.0 t_1)))
            (if (<= t_4 10.0) t_5 t_2)))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = fmax(fabs(one_es), fabs(sa));
	double t_2 = atan((((-ca * sin(lamt)) / t_1) / t_0));
	double t_3 = ca * sin(lamt);
	double t_4 = (tan(lamdp) * cos(lamt)) - t_3;
	double t_5 = atan((1.0 * (tan(lamdp) / (t_1 * t_0))));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = t_2;
	} else if (t_4 <= -0.09) {
		tmp = t_5;
	} else if (t_4 <= 1e-172) {
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_1))));
	} else if (t_4 <= 2e-42) {
		tmp = atan(((t_3 / t_0) * (-1.0 / t_1)));
	} else if (t_4 <= 10.0) {
		tmp = t_5;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_2 = Math.atan((((-ca * Math.sin(lamt)) / t_1) / t_0));
	double t_3 = ca * Math.sin(lamt);
	double t_4 = (Math.tan(lamdp) * Math.cos(lamt)) - t_3;
	double t_5 = Math.atan((1.0 * (Math.tan(lamdp) / (t_1 * t_0))));
	double tmp;
	if (t_4 <= -50000.0) {
		tmp = t_2;
	} else if (t_4 <= -0.09) {
		tmp = t_5;
	} else if (t_4 <= 1e-172) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (t_0 * t_1))));
	} else if (t_4 <= 2e-42) {
		tmp = Math.atan(((t_3 / t_0) * (-1.0 / t_1)));
	} else if (t_4 <= 10.0) {
		tmp = t_5;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = fmax(math.fabs(one_es), math.fabs(sa))
	t_2 = math.atan((((-ca * math.sin(lamt)) / t_1) / t_0))
	t_3 = ca * math.sin(lamt)
	t_4 = (math.tan(lamdp) * math.cos(lamt)) - t_3
	t_5 = math.atan((1.0 * (math.tan(lamdp) / (t_1 * t_0))))
	tmp = 0
	if t_4 <= -50000.0:
		tmp = t_2
	elif t_4 <= -0.09:
		tmp = t_5
	elif t_4 <= 1e-172:
		tmp = math.atan((math.cos(lamt) * (lamdp / (t_0 * t_1))))
	elif t_4 <= 2e-42:
		tmp = math.atan(((t_3 / t_0) * (-1.0 / t_1)))
	elif t_4 <= 10.0:
		tmp = t_5
	else:
		tmp = t_2
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = fmax(abs(one_es), abs(sa))
	t_2 = atan(Float64(Float64(Float64(Float64(-ca) * sin(lamt)) / t_1) / t_0))
	t_3 = Float64(ca * sin(lamt))
	t_4 = Float64(Float64(tan(lamdp) * cos(lamt)) - t_3)
	t_5 = atan(Float64(1.0 * Float64(tan(lamdp) / Float64(t_1 * t_0))))
	tmp = 0.0
	if (t_4 <= -50000.0)
		tmp = t_2;
	elseif (t_4 <= -0.09)
		tmp = t_5;
	elseif (t_4 <= 1e-172)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(t_0 * t_1))));
	elseif (t_4 <= 2e-42)
		tmp = atan(Float64(Float64(t_3 / t_0) * Float64(-1.0 / t_1)));
	elseif (t_4 <= 10.0)
		tmp = t_5;
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = max(abs(one_es), abs(sa));
	t_2 = atan((((-ca * sin(lamt)) / t_1) / t_0));
	t_3 = ca * sin(lamt);
	t_4 = (tan(lamdp) * cos(lamt)) - t_3;
	t_5 = atan((1.0 * (tan(lamdp) / (t_1 * t_0))));
	tmp = 0.0;
	if (t_4 <= -50000.0)
		tmp = t_2;
	elseif (t_4 <= -0.09)
		tmp = t_5;
	elseif (t_4 <= 1e-172)
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_1))));
	elseif (t_4 <= 2e-42)
		tmp = atan(((t_3 / t_0) * (-1.0 / t_1)));
	elseif (t_4 <= 10.0)
		tmp = t_5;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[N[(1.0 * N[(N[Tan[lamdp], $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -50000.0], t$95$2, If[LessEqual[t$95$4, -0.09], t$95$5, If[LessEqual[t$95$4, 1e-172], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e-42], N[ArcTan[N[(N[(t$95$3 / t$95$0), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 10.0], t$95$5, t$95$2]]]]]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4 or 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -0.089999999999999997 or 2.0000000000000001e-42 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(1 \cdot \frac{\color{blue}{\tan lamdp}}{sa \cdot one\_es}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 53.3%

      \[\leadsto \tan^{-1} \left(1 \cdot \frac{\color{blue}{\tan lamdp}}{sa \cdot one\_es}\right) \]

   if -0.089999999999999997 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 1e-172

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

   if 1e-172 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 2.0000000000000001e-42

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}}{sa}\right)} \]

      4. frac-2neg N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)}{\mathsf{neg}\left(sa\right)}\right)} \]

      5. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right)} \]

      7. distribute-neg-frac N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)\right)}{one\_es}} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)\right)}{one\_es}} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      9. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\right)}\right)}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt - \tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt - \tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\sin lamt \cdot ca} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\sin lamt \cdot ca} - \tan lamdp \cdot \cos lamt}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\tan lamdp \cdot \cos lamt}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\cos lamt \cdot \tan lamdp}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot ca - \color{blue}{\cos lamt \cdot \tan lamdp}}{one\_es} \cdot \frac{1}{\mathsf{neg}\left(sa\right)}\right) \]

   3. Applied rewrites 97.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\sin lamt \cdot ca - \cos lamt \cdot \tan lamdp}{one\_es} \cdot \frac{-1}{sa}\right)} \]

   4. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt}}{one\_es} \cdot \frac{-1}{sa}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{ca \cdot \color{blue}{\sin lamt}}{one\_es} \cdot \frac{-1}{sa}\right) \]

      2. lower-sin.f64 64.1%

         \[\leadsto \tan^{-1} \left(\frac{ca \cdot \sin lamt}{one\_es} \cdot \frac{-1}{sa}\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{ca \cdot \sin lamt}}{one\_es} \cdot \frac{-1}{sa}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \left(-ca\right) \cdot \sin lamt\\ t_2 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_3 := \tan^{-1} \left(1 \cdot \frac{\tan lamdp}{t\_2 \cdot t\_0}\right)\\ t_4 := \tan^{-1} \left(\frac{\frac{t\_1}{t\_2}}{t\_0}\right)\\ t_5 := \tan lamdp \cdot \cos lamt - ca \cdot \sin lamt\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -50000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq -0.09:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 10^{-172}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{t\_0 \cdot t\_2}\right)\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_1}{t\_0}}{t\_2}\right)\\ \mathbf{elif}\;t\_5 \leq 10:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (* (- ca) (sin lamt)))
       (t_2 (fmax (fabs one_es) (fabs sa)))
       (t_3 (atan (* 1.0 (/ (tan lamdp) (* t_2 t_0)))))
       (t_4 (atan (/ (/ t_1 t_2) t_0)))
       (t_5 (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt)))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= t_5 -50000.0)
      t_4
      (if (<= t_5 -0.09)
        t_3
        (if (<= t_5 1e-172)
          (atan (* (cos lamt) (/ lamdp (* t_0 t_2))))
          (if (<= t_5 2e-42)
            (atan (/ (/ t_1 t_0) t_2))
            (if (<= t_5 10.0) t_3 t_4)))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = -ca * sin(lamt);
	double t_2 = fmax(fabs(one_es), fabs(sa));
	double t_3 = atan((1.0 * (tan(lamdp) / (t_2 * t_0))));
	double t_4 = atan(((t_1 / t_2) / t_0));
	double t_5 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	double tmp;
	if (t_5 <= -50000.0) {
		tmp = t_4;
	} else if (t_5 <= -0.09) {
		tmp = t_3;
	} else if (t_5 <= 1e-172) {
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_2))));
	} else if (t_5 <= 2e-42) {
		tmp = atan(((t_1 / t_0) / t_2));
	} else if (t_5 <= 10.0) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = -ca * Math.sin(lamt);
	double t_2 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_3 = Math.atan((1.0 * (Math.tan(lamdp) / (t_2 * t_0))));
	double t_4 = Math.atan(((t_1 / t_2) / t_0));
	double t_5 = (Math.tan(lamdp) * Math.cos(lamt)) - (ca * Math.sin(lamt));
	double tmp;
	if (t_5 <= -50000.0) {
		tmp = t_4;
	} else if (t_5 <= -0.09) {
		tmp = t_3;
	} else if (t_5 <= 1e-172) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (t_0 * t_2))));
	} else if (t_5 <= 2e-42) {
		tmp = Math.atan(((t_1 / t_0) / t_2));
	} else if (t_5 <= 10.0) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = -ca * math.sin(lamt)
	t_2 = fmax(math.fabs(one_es), math.fabs(sa))
	t_3 = math.atan((1.0 * (math.tan(lamdp) / (t_2 * t_0))))
	t_4 = math.atan(((t_1 / t_2) / t_0))
	t_5 = (math.tan(lamdp) * math.cos(lamt)) - (ca * math.sin(lamt))
	tmp = 0
	if t_5 <= -50000.0:
		tmp = t_4
	elif t_5 <= -0.09:
		tmp = t_3
	elif t_5 <= 1e-172:
		tmp = math.atan((math.cos(lamt) * (lamdp / (t_0 * t_2))))
	elif t_5 <= 2e-42:
		tmp = math.atan(((t_1 / t_0) / t_2))
	elif t_5 <= 10.0:
		tmp = t_3
	else:
		tmp = t_4
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = Float64(Float64(-ca) * sin(lamt))
	t_2 = fmax(abs(one_es), abs(sa))
	t_3 = atan(Float64(1.0 * Float64(tan(lamdp) / Float64(t_2 * t_0))))
	t_4 = atan(Float64(Float64(t_1 / t_2) / t_0))
	t_5 = Float64(Float64(tan(lamdp) * cos(lamt)) - Float64(ca * sin(lamt)))
	tmp = 0.0
	if (t_5 <= -50000.0)
		tmp = t_4;
	elseif (t_5 <= -0.09)
		tmp = t_3;
	elseif (t_5 <= 1e-172)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(t_0 * t_2))));
	elseif (t_5 <= 2e-42)
		tmp = atan(Float64(Float64(t_1 / t_0) / t_2));
	elseif (t_5 <= 10.0)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = -ca * sin(lamt);
	t_2 = max(abs(one_es), abs(sa));
	t_3 = atan((1.0 * (tan(lamdp) / (t_2 * t_0))));
	t_4 = atan(((t_1 / t_2) / t_0));
	t_5 = (tan(lamdp) * cos(lamt)) - (ca * sin(lamt));
	tmp = 0.0;
	if (t_5 <= -50000.0)
		tmp = t_4;
	elseif (t_5 <= -0.09)
		tmp = t_3;
	elseif (t_5 <= 1e-172)
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_2))));
	elseif (t_5 <= 2e-42)
		tmp = atan(((t_1 / t_0) / t_2));
	elseif (t_5 <= 10.0)
		tmp = t_3;
	else
		tmp = t_4;
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(1.0 * N[(N[Tan[lamdp], $MachinePrecision] / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Tan[lamdp], $MachinePrecision] * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] - N[(ca * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, -50000.0], t$95$4, If[LessEqual[t$95$5, -0.09], t$95$3, If[LessEqual[t$95$5, 1e-172], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 2e-42], N[ArcTan[N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 10.0], t$95$3, t$95$4]]]]]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -5e4 or 10 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt)))

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -5e4 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < -0.089999999999999997 or 2.0000000000000001e-42 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 10

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(1 \cdot \frac{\color{blue}{\tan lamdp}}{sa \cdot one\_es}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 53.3%

      \[\leadsto \tan^{-1} \left(1 \cdot \frac{\color{blue}{\tan lamdp}}{sa \cdot one\_es}\right) \]

   if -0.089999999999999997 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 1e-172

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

   if 1e-172 < (-.f64 (*.f64 (tan.f64 lamdp) (cos.f64 lamt)) (*.f64 ca (sin.f64 lamt))) < 2.0000000000000001e-42

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      5. lower-/.f64 64.1%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}{sa}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es}}{sa}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es}}{sa}\right) \]

      8. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}}{one\_es}}{sa}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}}{one\_es}}{sa}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}}{one\_es}}{sa}\right) \]

      11. lower-neg.f64 64.1%

          \[\leadsto \tan^{-1} \left(\frac{\frac{\left(-ca\right) \cdot \sin \color{blue}{lamt}}{one\_es}}{sa}\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}{sa}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(-ca\right) \cdot \sin lamt\\ t_1 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_2 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;ca \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{t\_1}}{t\_2}\right)\\ \mathbf{elif}\;ca \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{t\_2 \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{t\_2}}{t\_1}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (- ca) (sin lamt)))
       (t_1 (fmax (fabs one_es) (fabs sa)))
       (t_2 (fmin (fabs one_es) (fabs sa))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= ca -3.9e-53)
      (atan (/ (/ t_0 t_1) t_2))
      (if (<= ca 5.2e-71)
        (atan (* (cos lamt) (/ lamdp (* t_2 t_1))))
        (atan (/ (/ t_0 t_2) t_1))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * sin(lamt);
	double t_1 = fmax(fabs(one_es), fabs(sa));
	double t_2 = fmin(fabs(one_es), fabs(sa));
	double tmp;
	if (ca <= -3.9e-53) {
		tmp = atan(((t_0 / t_1) / t_2));
	} else if (ca <= 5.2e-71) {
		tmp = atan((cos(lamt) * (lamdp / (t_2 * t_1))));
	} else {
		tmp = atan(((t_0 / t_2) / t_1));
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = -ca * Math.sin(lamt);
	double t_1 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_2 = fmin(Math.abs(one_es), Math.abs(sa));
	double tmp;
	if (ca <= -3.9e-53) {
		tmp = Math.atan(((t_0 / t_1) / t_2));
	} else if (ca <= 5.2e-71) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (t_2 * t_1))));
	} else {
		tmp = Math.atan(((t_0 / t_2) / t_1));
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = -ca * math.sin(lamt)
	t_1 = fmax(math.fabs(one_es), math.fabs(sa))
	t_2 = fmin(math.fabs(one_es), math.fabs(sa))
	tmp = 0
	if ca <= -3.9e-53:
		tmp = math.atan(((t_0 / t_1) / t_2))
	elif ca <= 5.2e-71:
		tmp = math.atan((math.cos(lamt) * (lamdp / (t_2 * t_1))))
	else:
		tmp = math.atan(((t_0 / t_2) / t_1))
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = Float64(Float64(-ca) * sin(lamt))
	t_1 = fmax(abs(one_es), abs(sa))
	t_2 = fmin(abs(one_es), abs(sa))
	tmp = 0.0
	if (ca <= -3.9e-53)
		tmp = atan(Float64(Float64(t_0 / t_1) / t_2));
	elseif (ca <= 5.2e-71)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(t_2 * t_1))));
	else
		tmp = atan(Float64(Float64(t_0 / t_2) / t_1));
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = -ca * sin(lamt);
	t_1 = max(abs(one_es), abs(sa));
	t_2 = min(abs(one_es), abs(sa));
	tmp = 0.0;
	if (ca <= -3.9e-53)
		tmp = atan(((t_0 / t_1) / t_2));
	elseif (ca <= 5.2e-71)
		tmp = atan((cos(lamt) * (lamdp / (t_2 * t_1))));
	else
		tmp = atan(((t_0 / t_2) / t_1));
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[ca, -3.9e-53], N[ArcTan[N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[ca, 5.2e-71], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$0 / t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ca < -3.9000000000000002e-53

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-1 \cdot \left(ca \cdot \sin lamt\right)\right) \cdot \frac{1}{one\_es \cdot sa}\right)} \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{one\_es \cdot sa} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right)} \]

      5. lower-/.f64 61.9%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{one\_es \cdot sa}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      8. lower-*.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{sa \cdot one\_es}} \cdot \left(-1 \cdot \left(ca \cdot \sin lamt\right)\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}\right)\right) \]

      14. lower-neg.f64 61.9%

          \[\leadsto \tan^{-1} \left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin \color{blue}{lamt}\right)\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{sa \cdot one\_es} \cdot \left(\left(-ca\right) \cdot \sin lamt\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \frac{1}{sa \cdot one\_es}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(\left(-ca\right) \cdot \sin lamt\right) \cdot \color{blue}{\frac{1}{sa \cdot one\_es}}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-ca\right) \cdot \sin lamt}{sa \cdot one\_es}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-ca\right) \cdot \sin lamt}{\color{blue}{sa \cdot one\_es}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

      8. lower-/.f64 64.2%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}}{one\_es}\right) \]

   8. Applied rewrites 64.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{sa}}{one\_es}\right)} \]

   if -3.9000000000000002e-53 < ca < 5.1999999999999997e-71

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

   if 5.1999999999999997e-71 < ca

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      5. lower-/.f64 64.1%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}{sa}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es}}{sa}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es}}{sa}\right) \]

      8. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}}{one\_es}}{sa}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}}{one\_es}}{sa}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}}{one\_es}}{sa}\right) \]

      11. lower-neg.f64 64.1%

          \[\leadsto \tan^{-1} \left(\frac{\frac{\left(-ca\right) \cdot \sin \color{blue}{lamt}}{one\_es}}{sa}\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}{sa}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_1 := \mathsf{max}\left(\left|one\_es\right|, \left|sa\right|\right)\\ t_2 := \tan^{-1} \left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{t\_0}}{t\_1}\right)\\ \mathsf{copysign}\left(1, one\_es\right) \cdot \left(\mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;ca \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ca \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{t\_0 \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fabs one_es) (fabs sa)))
       (t_1 (fmax (fabs one_es) (fabs sa)))
       (t_2 (atan (/ (/ (* (- ca) (sin lamt)) t_0) t_1))))
  (*
   (copysign 1.0 one_es)
   (*
    (copysign 1.0 sa)
    (if (<= ca -2.3e-53)
      t_2
      (if (<= ca 5.2e-71)
        (atan (* (cos lamt) (/ lamdp (* t_0 t_1))))
        t_2))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(fabs(one_es), fabs(sa));
	double t_1 = fmax(fabs(one_es), fabs(sa));
	double t_2 = atan((((-ca * sin(lamt)) / t_0) / t_1));
	double tmp;
	if (ca <= -2.3e-53) {
		tmp = t_2;
	} else if (ca <= 5.2e-71) {
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_1))));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, one_es) * (copysign(1.0, sa) * tmp);
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = fmin(Math.abs(one_es), Math.abs(sa));
	double t_1 = fmax(Math.abs(one_es), Math.abs(sa));
	double t_2 = Math.atan((((-ca * Math.sin(lamt)) / t_0) / t_1));
	double tmp;
	if (ca <= -2.3e-53) {
		tmp = t_2;
	} else if (ca <= 5.2e-71) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (t_0 * t_1))));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, one_es) * (Math.copySign(1.0, sa) * tmp);
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = fmin(math.fabs(one_es), math.fabs(sa))
	t_1 = fmax(math.fabs(one_es), math.fabs(sa))
	t_2 = math.atan((((-ca * math.sin(lamt)) / t_0) / t_1))
	tmp = 0
	if ca <= -2.3e-53:
		tmp = t_2
	elif ca <= 5.2e-71:
		tmp = math.atan((math.cos(lamt) * (lamdp / (t_0 * t_1))))
	else:
		tmp = t_2
	return math.copysign(1.0, one_es) * (math.copysign(1.0, sa) * tmp)

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = fmin(abs(one_es), abs(sa))
	t_1 = fmax(abs(one_es), abs(sa))
	t_2 = atan(Float64(Float64(Float64(Float64(-ca) * sin(lamt)) / t_0) / t_1))
	tmp = 0.0
	if (ca <= -2.3e-53)
		tmp = t_2;
	elseif (ca <= 5.2e-71)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(t_0 * t_1))));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, one_es) * Float64(copysign(1.0, sa) * tmp))
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = min(abs(one_es), abs(sa));
	t_1 = max(abs(one_es), abs(sa));
	t_2 = atan((((-ca * sin(lamt)) / t_0) / t_1));
	tmp = 0.0;
	if (ca <= -2.3e-53)
		tmp = t_2;
	elseif (ca <= 5.2e-71)
		tmp = atan((cos(lamt) * (lamdp / (t_0 * t_1))));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(one_es) * abs(1.0)) * ((sign(sa) * abs(1.0)) * tmp);
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[Min[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[one$95$es], $MachinePrecision], N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[((-ca) * N[Sin[lamt], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[one$95$es]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[ca, -2.3e-53], t$95$2, If[LessEqual[ca, 5.2e-71], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ca < -2.3000000000000001e-53 or 5.1999999999999997e-71 < ca

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}{sa}\right)} \]

      5. lower-/.f64 64.1%

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es}}}{sa}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es}}{sa}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es}}{sa}\right) \]

      8. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}}{one\_es}}{sa}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}}{one\_es}}{sa}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \color{blue}{\sin lamt}}{one\_es}}{sa}\right) \]

      11. lower-neg.f64 64.1%

          \[\leadsto \tan^{-1} \left(\frac{\frac{\left(-ca\right) \cdot \sin \color{blue}{lamt}}{one\_es}}{sa}\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(-ca\right) \cdot \sin lamt}{one\_es}}{sa}\right)} \]

   if -2.3000000000000001e-53 < ca < 5.1999999999999997e-71

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 71.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{\sin lamt \cdot \left(-ca\right)}{one\_es \cdot sa}\right)\\ \mathbf{if}\;ca \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;ca \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (atan (/ (* (sin lamt) (- ca)) (* one_es sa)))))
  (if (<= ca -2.3e-53)
    t_0
    (if (<= ca 5.2e-71)
      (atan (* (cos lamt) (/ lamdp (* one_es sa))))
      t_0))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = atan(((sin(lamt) * -ca) / (one_es * sa)));
	double tmp;
	if (ca <= -2.3e-53) {
		tmp = t_0;
	} else if (ca <= 5.2e-71) {
		tmp = atan((cos(lamt) * (lamdp / (one_es * sa))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan(((sin(lamt) * -ca) / (one_es * sa)))
    if (ca <= (-2.3d-53)) then
        tmp = t_0
    else if (ca <= 5.2d-71) then
        tmp = atan((cos(lamt) * (lamdp / (one_es * sa))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = Math.atan(((Math.sin(lamt) * -ca) / (one_es * sa)));
	double tmp;
	if (ca <= -2.3e-53) {
		tmp = t_0;
	} else if (ca <= 5.2e-71) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (one_es * sa))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = math.atan(((math.sin(lamt) * -ca) / (one_es * sa)))
	tmp = 0
	if ca <= -2.3e-53:
		tmp = t_0
	elif ca <= 5.2e-71:
		tmp = math.atan((math.cos(lamt) * (lamdp / (one_es * sa))))
	else:
		tmp = t_0
	return tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = atan(Float64(Float64(sin(lamt) * Float64(-ca)) / Float64(one_es * sa)))
	tmp = 0.0
	if (ca <= -2.3e-53)
		tmp = t_0;
	elseif (ca <= 5.2e-71)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(one_es * sa))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = atan(((sin(lamt) * -ca) / (one_es * sa)));
	tmp = 0.0;
	if (ca <= -2.3e-53)
		tmp = t_0;
	elseif (ca <= 5.2e-71)
		tmp = atan((cos(lamt) * (lamdp / (one_es * sa))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[ArcTan[N[(N[(N[Sin[lamt], $MachinePrecision] * (-ca)), $MachinePrecision] / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ca, -2.3e-53], t$95$0, If[LessEqual[ca, 5.2e-71], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(one$95$es * sa), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	LET t_0 = (atan((((sin(lamt)) * (- ca)) / (one_es * sa)))) IN
		LET tmp_1 = IF (ca <= (51999999999999997200854947069912431079022772637829376591480882318659722597362398162754541116632406650803952489103832099832937233245082513237751020631650455937387550546559798238244187729151235544122755527496337890625e-285)) THEN (atan(((cos(lamt)) * (lamdp / (one_es * sa))))) ELSE t_0 ENDIF IN
		LET tmp = IF (ca <= (-230000000000000012879349904392203014443046892407242736861512823798692403166447348251053572979275094381800949352384342051152362479816797236065184506514924578368663787841796875e-226)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if ca < -2.3000000000000001e-53 or 5.1999999999999997e-71 < ca

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-1 \cdot ca\right) \cdot \color{blue}{\sin lamt}}{one\_es \cdot sa}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(ca\right)\right) \cdot \sin \color{blue}{lamt}}{one\_es \cdot sa}\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot \color{blue}{\left(\mathsf{neg}\left(ca\right)\right)}}{one\_es \cdot sa}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot \color{blue}{\left(\mathsf{neg}\left(ca\right)\right)}}{one\_es \cdot sa}\right) \]

      7. lower-neg.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot \left(-ca\right)}{one\_es \cdot sa}\right) \]

   6. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\sin lamt \cdot \color{blue}{\left(-ca\right)}}{one\_es \cdot sa}\right) \]

   if -2.3000000000000001e-53 < ca < 5.1999999999999997e-71

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 56.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt\\ t_1 := \mathsf{max}\left(one\_es, \left|sa\right|\right)\\ t_2 := \mathsf{min}\left(one\_es, \left|sa\right|\right)\\ \mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;ca \leq -4.8 \cdot 10^{+63}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{t\_1}}{t\_2}\right)\\ \mathbf{elif}\;ca \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{t\_2 \cdot t\_1}\right)\\ \mathbf{elif}\;ca \leq 1.2 \cdot 10^{+39}:\\ \;\;\;\;\tan^{-1} \left(\frac{-ca \cdot lamt}{t\_1 \cdot t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{t\_0}{t\_2} \cdot \frac{1}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0
        (* (- (* (* (* ca lamt) lamt) 0.16666666666666666) ca) lamt))
       (t_1 (fmax one_es (fabs sa)))
       (t_2 (fmin one_es (fabs sa))))
  (*
   (copysign 1.0 sa)
   (if (<= ca -4.8e+63)
     (atan (/ (/ t_0 t_1) t_2))
     (if (<= ca 5.2e-71)
       (atan (* (cos lamt) (/ lamdp (* t_2 t_1))))
       (if (<= ca 1.2e+39)
         (atan (/ (- (* ca lamt)) (* t_1 t_2)))
         (atan (* (/ t_0 t_2) (/ 1.0 t_1)))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt;
	double t_1 = fmax(one_es, fabs(sa));
	double t_2 = fmin(one_es, fabs(sa));
	double tmp;
	if (ca <= -4.8e+63) {
		tmp = atan(((t_0 / t_1) / t_2));
	} else if (ca <= 5.2e-71) {
		tmp = atan((cos(lamt) * (lamdp / (t_2 * t_1))));
	} else if (ca <= 1.2e+39) {
		tmp = atan((-(ca * lamt) / (t_1 * t_2)));
	} else {
		tmp = atan(((t_0 / t_2) * (1.0 / t_1)));
	}
	return copysign(1.0, sa) * tmp;
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt;
	double t_1 = fmax(one_es, Math.abs(sa));
	double t_2 = fmin(one_es, Math.abs(sa));
	double tmp;
	if (ca <= -4.8e+63) {
		tmp = Math.atan(((t_0 / t_1) / t_2));
	} else if (ca <= 5.2e-71) {
		tmp = Math.atan((Math.cos(lamt) * (lamdp / (t_2 * t_1))));
	} else if (ca <= 1.2e+39) {
		tmp = Math.atan((-(ca * lamt) / (t_1 * t_2)));
	} else {
		tmp = Math.atan(((t_0 / t_2) * (1.0 / t_1)));
	}
	return Math.copySign(1.0, sa) * tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt
	t_1 = fmax(one_es, math.fabs(sa))
	t_2 = fmin(one_es, math.fabs(sa))
	tmp = 0
	if ca <= -4.8e+63:
		tmp = math.atan(((t_0 / t_1) / t_2))
	elif ca <= 5.2e-71:
		tmp = math.atan((math.cos(lamt) * (lamdp / (t_2 * t_1))))
	elif ca <= 1.2e+39:
		tmp = math.atan((-(ca * lamt) / (t_1 * t_2)))
	else:
		tmp = math.atan(((t_0 / t_2) * (1.0 / t_1)))
	return math.copysign(1.0, sa) * tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = Float64(Float64(Float64(Float64(Float64(ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt)
	t_1 = fmax(one_es, abs(sa))
	t_2 = fmin(one_es, abs(sa))
	tmp = 0.0
	if (ca <= -4.8e+63)
		tmp = atan(Float64(Float64(t_0 / t_1) / t_2));
	elseif (ca <= 5.2e-71)
		tmp = atan(Float64(cos(lamt) * Float64(lamdp / Float64(t_2 * t_1))));
	elseif (ca <= 1.2e+39)
		tmp = atan(Float64(Float64(-Float64(ca * lamt)) / Float64(t_1 * t_2)));
	else
		tmp = atan(Float64(Float64(t_0 / t_2) * Float64(1.0 / t_1)));
	end
	return Float64(copysign(1.0, sa) * tmp)
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt;
	t_1 = max(one_es, abs(sa));
	t_2 = min(one_es, abs(sa));
	tmp = 0.0;
	if (ca <= -4.8e+63)
		tmp = atan(((t_0 / t_1) / t_2));
	elseif (ca <= 5.2e-71)
		tmp = atan((cos(lamt) * (lamdp / (t_2 * t_1))));
	elseif (ca <= 1.2e+39)
		tmp = atan((-(ca * lamt) / (t_1 * t_2)));
	else
		tmp = atan(((t_0 / t_2) * (1.0 / t_1)));
	end
	tmp_2 = (sign(sa) * abs(1.0)) * tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[(N[(N[(N[(N[(ca * lamt), $MachinePrecision] * lamt), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - ca), $MachinePrecision] * lamt), $MachinePrecision]}, Block[{t$95$1 = N[Max[one$95$es, N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[one$95$es, N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[ca, -4.8e+63], N[ArcTan[N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[ca, 5.2e-71], N[ArcTan[N[(N[Cos[lamt], $MachinePrecision] * N[(lamdp / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ca, 1.2e+39], N[ArcTan[N[((-N[(ca * lamt), $MachinePrecision]) / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$0 / t$95$2), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if ca < -4.8e63

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\left(-1 \cdot ca + \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \left(-1 \cdot ca + \color{blue}{\frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)}\right)}{one\_es \cdot sa}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      5. lower-pow.f64 43.1%

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{sa \cdot one\_es}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{sa}}{one\_es}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{sa}}{one\_es}\right)} \]

   9. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{sa}}{one\_es}\right)} \]

   if -4.8e63 < ca < 5.1999999999999997e-71

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \color{blue}{\frac{\sin lamdp}{sa \cdot \cos lamdp}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{sa \cdot \color{blue}{\cos lamdp}}\right) \]

      6. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\sin lamdp}{\cos lamdp \cdot \color{blue}{sa}}\right) \]

      7. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{\color{blue}{sa}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\frac{\sin lamdp}{\cos lamdp}}{sa}\right) \]

      10. tan-quot N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt}{one\_es} \cdot \frac{\tan lamdp}{sa}\right) \]

      12. times-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{one\_es \cdot sa}}\right) \]

      16. lower-/.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      18. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

      19. lower-*.f64 63.7%

          \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{\tan lamdp}{sa \cdot \color{blue}{one\_es}}\right) \]

   6. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \color{blue}{\frac{\tan lamdp}{sa \cdot one\_es}}\right) \]

   7. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{one\_es \cdot sa}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\cos lamt \cdot \frac{lamdp}{\color{blue}{one\_es \cdot sa}}\right) \]

   if 5.1999999999999997e-71 < ca < 1.2e39

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot lamt\right)}}{one\_es \cdot sa}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

      3. lower-neg.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{one\_es \cdot sa}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(one\_es \cdot sa\right)\right)}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(*-commutative, \left(sa \cdot one\_es\right)\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(sa \cdot one\_es\right)\right)}\right) \]

   9. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)} \]

   if 1.2e39 < ca

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\left(-1 \cdot ca + \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \left(-1 \cdot ca + \color{blue}{\frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)}\right)}{one\_es \cdot sa}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      5. lower-pow.f64 43.1%

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es}}{sa}\right)} \]

      4. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es} \cdot \frac{1}{sa}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es} \cdot \frac{1}{sa}\right)} \]

   9. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{one\_es} \cdot \frac{1}{sa}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt\\ t_1 := \mathsf{max}\left(one\_es, \left|sa\right|\right)\\ t_2 := \mathsf{min}\left(one\_es, \left|sa\right|\right)\\ \mathsf{copysign}\left(1, sa\right) \cdot \begin{array}{l} \mathbf{if}\;ca \leq -4.8 \cdot 10^{+63}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{t\_0}{t\_1}}{t\_2}\right)\\ \mathbf{elif}\;ca \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{t\_2 \cdot t\_1}\right)\\ \mathbf{elif}\;ca \leq 1.2 \cdot 10^{+39}:\\ \;\;\;\;\tan^{-1} \left(\frac{-ca \cdot lamt}{t\_1 \cdot t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{t\_0}{t\_2} \cdot \frac{1}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (let* ((t_0
        (* (- (* (* (* ca lamt) lamt) 0.16666666666666666) ca) lamt))
       (t_1 (fmax one_es (fabs sa)))
       (t_2 (fmin one_es (fabs sa))))
  (*
   (copysign 1.0 sa)
   (if (<= ca -4.8e+63)
     (atan (/ (/ t_0 t_1) t_2))
     (if (<= ca 5.2e-71)
       (atan (/ (* lamdp (cos lamt)) (* t_2 t_1)))
       (if (<= ca 1.2e+39)
         (atan (/ (- (* ca lamt)) (* t_1 t_2)))
         (atan (* (/ t_0 t_2) (/ 1.0 t_1)))))))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt;
	double t_1 = fmax(one_es, fabs(sa));
	double t_2 = fmin(one_es, fabs(sa));
	double tmp;
	if (ca <= -4.8e+63) {
		tmp = atan(((t_0 / t_1) / t_2));
	} else if (ca <= 5.2e-71) {
		tmp = atan(((lamdp * cos(lamt)) / (t_2 * t_1)));
	} else if (ca <= 1.2e+39) {
		tmp = atan((-(ca * lamt) / (t_1 * t_2)));
	} else {
		tmp = atan(((t_0 / t_2) * (1.0 / t_1)));
	}
	return copysign(1.0, sa) * tmp;
}

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt;
	double t_1 = fmax(one_es, Math.abs(sa));
	double t_2 = fmin(one_es, Math.abs(sa));
	double tmp;
	if (ca <= -4.8e+63) {
		tmp = Math.atan(((t_0 / t_1) / t_2));
	} else if (ca <= 5.2e-71) {
		tmp = Math.atan(((lamdp * Math.cos(lamt)) / (t_2 * t_1)));
	} else if (ca <= 1.2e+39) {
		tmp = Math.atan((-(ca * lamt) / (t_1 * t_2)));
	} else {
		tmp = Math.atan(((t_0 / t_2) * (1.0 / t_1)));
	}
	return Math.copySign(1.0, sa) * tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt
	t_1 = fmax(one_es, math.fabs(sa))
	t_2 = fmin(one_es, math.fabs(sa))
	tmp = 0
	if ca <= -4.8e+63:
		tmp = math.atan(((t_0 / t_1) / t_2))
	elif ca <= 5.2e-71:
		tmp = math.atan(((lamdp * math.cos(lamt)) / (t_2 * t_1)))
	elif ca <= 1.2e+39:
		tmp = math.atan((-(ca * lamt) / (t_1 * t_2)))
	else:
		tmp = math.atan(((t_0 / t_2) * (1.0 / t_1)))
	return math.copysign(1.0, sa) * tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	t_0 = Float64(Float64(Float64(Float64(Float64(ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt)
	t_1 = fmax(one_es, abs(sa))
	t_2 = fmin(one_es, abs(sa))
	tmp = 0.0
	if (ca <= -4.8e+63)
		tmp = atan(Float64(Float64(t_0 / t_1) / t_2));
	elseif (ca <= 5.2e-71)
		tmp = atan(Float64(Float64(lamdp * cos(lamt)) / Float64(t_2 * t_1)));
	elseif (ca <= 1.2e+39)
		tmp = atan(Float64(Float64(-Float64(ca * lamt)) / Float64(t_1 * t_2)));
	else
		tmp = atan(Float64(Float64(t_0 / t_2) * Float64(1.0 / t_1)));
	end
	return Float64(copysign(1.0, sa) * tmp)
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	t_0 = ((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt;
	t_1 = max(one_es, abs(sa));
	t_2 = min(one_es, abs(sa));
	tmp = 0.0;
	if (ca <= -4.8e+63)
		tmp = atan(((t_0 / t_1) / t_2));
	elseif (ca <= 5.2e-71)
		tmp = atan(((lamdp * cos(lamt)) / (t_2 * t_1)));
	elseif (ca <= 1.2e+39)
		tmp = atan((-(ca * lamt) / (t_1 * t_2)));
	else
		tmp = atan(((t_0 / t_2) * (1.0 / t_1)));
	end
	tmp_2 = (sign(sa) * abs(1.0)) * tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := Block[{t$95$0 = N[(N[(N[(N[(N[(ca * lamt), $MachinePrecision] * lamt), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - ca), $MachinePrecision] * lamt), $MachinePrecision]}, Block[{t$95$1 = N[Max[one$95$es, N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[one$95$es, N[Abs[sa], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[sa]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[ca, -4.8e+63], N[ArcTan[N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[ca, 5.2e-71], N[ArcTan[N[(N[(lamdp * N[Cos[lamt], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ca, 1.2e+39], N[ArcTan[N[((-N[(ca * lamt), $MachinePrecision]) / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$0 / t$95$2), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if ca < -4.8e63

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\left(-1 \cdot ca + \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \left(-1 \cdot ca + \color{blue}{\frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)}\right)}{one\_es \cdot sa}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      5. lower-pow.f64 43.1%

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{sa \cdot one\_es}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{sa}}{one\_es}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{sa}}{one\_es}\right)} \]

   9. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{sa}}{one\_es}\right)} \]

   if -4.8e63 < ca < 5.1999999999999997e-71

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in ca around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{\color{blue}{one\_es} \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \color{blue}{\left(sa \cdot \cos lamdp\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \color{blue}{\cos lamdp}\right)}\right) \]

      7. lower-cos.f64 63.7%

         \[\leadsto \tan^{-1} \left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right) \]

   4. Applied rewrites 63.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\cos lamt \cdot \sin lamdp}{one\_es \cdot \left(sa \cdot \cos lamdp\right)}\right)} \]

   5. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{\color{blue}{one\_es \cdot sa}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{one\_es \cdot \color{blue}{sa}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{one\_es \cdot sa}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 45.7%

         \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 45.7%

      \[\leadsto \tan^{-1} \left(\frac{lamdp \cdot \cos lamt}{\color{blue}{one\_es \cdot sa}}\right) \]

   if 5.1999999999999997e-71 < ca < 1.2e39

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot lamt\right)}}{one\_es \cdot sa}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

      3. lower-neg.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{one\_es \cdot sa}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(one\_es \cdot sa\right)\right)}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(*-commutative, \left(sa \cdot one\_es\right)\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(sa \cdot one\_es\right)\right)}\right) \]

   9. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)} \]

   if 1.2e39 < ca

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\left(-1 \cdot ca + \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \left(-1 \cdot ca + \color{blue}{\frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)}\right)}{one\_es \cdot sa}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      5. lower-pow.f64 43.1%

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es}}{sa}\right)} \]

      4. mult-flip N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es} \cdot \frac{1}{sa}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es} \cdot \frac{1}{sa}\right)} \]

   9. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{one\_es} \cdot \frac{1}{sa}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;lamt \leq -4.2 \cdot 10^{+59}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{sa}}{one\_es}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)\\ \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (if (<= lamt -4.2e+59)
  (atan
   (/
    (/
     (* (- (* (* (* ca lamt) lamt) 0.16666666666666666) ca) lamt)
     sa)
    one_es))
  (atan (/ (- (* ca lamt)) (* sa one_es)))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double tmp;
	if (lamt <= -4.2e+59) {
		tmp = atan((((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / sa) / one_es));
	} else {
		tmp = atan((-(ca * lamt) / (sa * one_es)));
	}
	return tmp;
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    real(8) :: tmp
    if (lamt <= (-4.2d+59)) then
        tmp = atan((((((((ca * lamt) * lamt) * 0.16666666666666666d0) - ca) * lamt) / sa) / one_es))
    else
        tmp = atan((-(ca * lamt) / (sa * one_es)))
    end if
    code = tmp
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double tmp;
	if (lamt <= -4.2e+59) {
		tmp = Math.atan((((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / sa) / one_es));
	} else {
		tmp = Math.atan((-(ca * lamt) / (sa * one_es)));
	}
	return tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	tmp = 0
	if lamt <= -4.2e+59:
		tmp = math.atan((((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / sa) / one_es))
	else:
		tmp = math.atan((-(ca * lamt) / (sa * one_es)))
	return tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	tmp = 0.0
	if (lamt <= -4.2e+59)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(Float64(Float64(ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / sa) / one_es));
	else
		tmp = atan(Float64(Float64(-Float64(ca * lamt)) / Float64(sa * one_es)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	tmp = 0.0;
	if (lamt <= -4.2e+59)
		tmp = atan((((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / sa) / one_es));
	else
		tmp = atan((-(ca * lamt) / (sa * one_es)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := If[LessEqual[lamt, -4.2e+59], N[ArcTan[N[(N[(N[(N[(N[(N[(N[(ca * lamt), $MachinePrecision] * lamt), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - ca), $MachinePrecision] * lamt), $MachinePrecision] / sa), $MachinePrecision] / one$95$es), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[((-N[(ca * lamt), $MachinePrecision]) / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	LET tmp = IF (lamt <= (-419999999999999968038239129476388816187586133754418707824640)) THEN (atan((((((((ca * lamt) * lamt) * (1666666666666666574148081281236954964697360992431640625e-55)) - ca) * lamt) / sa) / one_es))) ELSE (atan(((- (ca * lamt)) / (sa * one_es)))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if lamt < -4.1999999999999997e59

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\left(-1 \cdot ca + \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \left(-1 \cdot ca + \color{blue}{\frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)}\right)}{one\_es \cdot sa}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      5. lower-pow.f64 43.1%

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{one\_es \cdot sa}}\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{\color{blue}{sa \cdot one\_es}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{sa}}{one\_es}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{sa}}{one\_es}\right)} \]

   9. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{sa}}{one\_es}\right)} \]

   if -4.1999999999999997e59 < lamt

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot lamt\right)}}{one\_es \cdot sa}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

      3. lower-neg.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{one\_es \cdot sa}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(one\_es \cdot sa\right)\right)}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(*-commutative, \left(sa \cdot one\_es\right)\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(sa \cdot one\_es\right)\right)}\right) \]

   9. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;lamt \leq -4.2 \cdot 10^{+59}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{sa \cdot one\_es}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)\\ \end{array} \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (if (<= lamt -4.2e+59)
  (atan
   (/
    (* (- (* (* (* ca lamt) lamt) 0.16666666666666666) ca) lamt)
    (* sa one_es)))
  (atan (/ (- (* ca lamt)) (* sa one_es)))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double tmp;
	if (lamt <= -4.2e+59) {
		tmp = atan(((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / (sa * one_es)));
	} else {
		tmp = atan((-(ca * lamt) / (sa * one_es)));
	}
	return tmp;
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    real(8) :: tmp
    if (lamt <= (-4.2d+59)) then
        tmp = atan(((((((ca * lamt) * lamt) * 0.16666666666666666d0) - ca) * lamt) / (sa * one_es)))
    else
        tmp = atan((-(ca * lamt) / (sa * one_es)))
    end if
    code = tmp
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	double tmp;
	if (lamt <= -4.2e+59) {
		tmp = Math.atan(((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / (sa * one_es)));
	} else {
		tmp = Math.atan((-(ca * lamt) / (sa * one_es)));
	}
	return tmp;
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	tmp = 0
	if lamt <= -4.2e+59:
		tmp = math.atan(((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / (sa * one_es)))
	else:
		tmp = math.atan((-(ca * lamt) / (sa * one_es)))
	return tmp

Julia

function code(lamdp, lamt, ca, one_es, sa)
	tmp = 0.0
	if (lamt <= -4.2e+59)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(Float64(ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / Float64(sa * one_es)));
	else
		tmp = atan(Float64(Float64(-Float64(ca * lamt)) / Float64(sa * one_es)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lamdp, lamt, ca, one_es, sa)
	tmp = 0.0;
	if (lamt <= -4.2e+59)
		tmp = atan(((((((ca * lamt) * lamt) * 0.16666666666666666) - ca) * lamt) / (sa * one_es)));
	else
		tmp = atan((-(ca * lamt) / (sa * one_es)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := If[LessEqual[lamt, -4.2e+59], N[ArcTan[N[(N[(N[(N[(N[(N[(ca * lamt), $MachinePrecision] * lamt), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - ca), $MachinePrecision] * lamt), $MachinePrecision] / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[((-N[(ca * lamt), $MachinePrecision]) / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	LET tmp = IF (lamt <= (-419999999999999968038239129476388816187586133754418707824640)) THEN (atan(((((((ca * lamt) * lamt) * (1666666666666666574148081281236954964697360992431640625e-55)) - ca) * lamt) / (sa * one_es)))) ELSE (atan(((- (ca * lamt)) / (sa * one_es)))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if lamt < -4.1999999999999997e59

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\left(-1 \cdot ca + \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \left(-1 \cdot ca + \color{blue}{\frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)}\right)}{one\_es \cdot sa}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, \frac{1}{6} \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

      5. lower-pow.f64 43.1%

         \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \left(\frac{lamt \cdot \color{blue}{\mathsf{fma}\left(-1, ca, 0.16666666666666666 \cdot \left(ca \cdot {lamt}^{2}\right)\right)}}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 43.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\left(\left(ca \cdot lamt\right) \cdot lamt\right) \cdot 0.16666666666666666 - ca\right) \cdot lamt}{sa \cdot one\_es}\right)} \]

   if -4.1999999999999997e59 < lamt

   1. Initial program 96.7%

      \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

   2. Taylor expanded in lamdp around 0

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

      3. lower-sin.f64 61.9%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

   4. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   5. Taylor expanded in lamt around 0

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

   7. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot lamt\right)}}{one\_es \cdot sa}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

      3. lower-neg.f64 44.6%

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{one\_es \cdot sa}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(one\_es \cdot sa\right)\right)}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(*-commutative, \left(sa \cdot one\_es\right)\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(sa \cdot one\_es\right)\right)}\right) \]

   9. Applied rewrites 44.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.9% accurate, 6.3× speedup

Math

\[\tan^{-1} \left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right) \]

FPCore

(FPCore (lamdp lamt ca one_es sa)
  :precision binary64
  :pre TRUE
  (atan (/ (- (* ca lamt)) (* sa one_es))))

C

double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return atan((-(ca * lamt) / (sa * one_es)));
}

Fortran

real(8) function code(lamdp, lamt, ca, one_es, sa)
use fmin_fmax_functions
    real(8), intent (in) :: lamdp
    real(8), intent (in) :: lamt
    real(8), intent (in) :: ca
    real(8), intent (in) :: one_es
    real(8), intent (in) :: sa
    code = atan((-(ca * lamt) / (sa * one_es)))
end function

Java

public static double code(double lamdp, double lamt, double ca, double one_es, double sa) {
	return Math.atan((-(ca * lamt) / (sa * one_es)));
}

Python

def code(lamdp, lamt, ca, one_es, sa):
	return math.atan((-(ca * lamt) / (sa * one_es)))

Julia

function code(lamdp, lamt, ca, one_es, sa)
	return atan(Float64(Float64(-Float64(ca * lamt)) / Float64(sa * one_es)))
end

MATLAB

function tmp = code(lamdp, lamt, ca, one_es, sa)
	tmp = atan((-(ca * lamt) / (sa * one_es)));
end

Wolfram

code[lamdp_, lamt_, ca_, one$95$es_, sa_] := N[ArcTan[N[((-N[(ca * lamt), $MachinePrecision]) / N[(sa * one$95$es), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

ReFlow

f(lamdp, lamt, ca, one_es, sa):
	lamdp in [-inf, +inf],
	lamt in [-inf, +inf],
	ca in [-inf, +inf],
	one_es in [-inf, +inf],
	sa in [-inf, +inf]
code: THEORY
BEGIN
f(lamdp, lamt, ca, one_es, sa: real): real =
	atan(((- (ca * lamt)) / (sa * one_es)))
END code

Derivation

1. Initial program 96.7%

   \[\tan^{-1} \left(\frac{\tan lamdp \cdot \cos lamt - ca \cdot \sin lamt}{one\_es \cdot sa}\right) \]

2. Taylor expanded in lamdp around 0

   \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{\sin lamt}\right)}{one\_es \cdot sa}\right) \]

   3. lower-sin.f64 61.9%

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \sin lamt\right)}{one\_es \cdot sa}\right) \]

4. Applied rewrites 61.9%

   \[\leadsto \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \left(ca \cdot \sin lamt\right)}}{one\_es \cdot sa}\right) \]

5. Taylor expanded in lamt around 0

   \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 44.6%

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

7. Applied rewrites 44.6%

   \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \left(ca \cdot \color{blue}{lamt}\right)}{one\_es \cdot sa}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(ca \cdot lamt\right)}}{one\_es \cdot sa}\right) \]

   2. mul-1-neg N/A

      \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(ca \cdot lamt\right)}{one\_es \cdot sa}\right) \]

   3. lower-neg.f64 44.6%

      \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{one\_es \cdot sa}\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(one\_es \cdot sa\right)\right)}\right) \]

   5. lower-neg.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(*-commutative, \left(sa \cdot one\_es\right)\right)}\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{-ca \cdot lamt}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(sa \cdot one\_es\right)\right)}\right) \]

9. Applied rewrites 44.6%

   \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-ca \cdot lamt}{sa \cdot one\_es}\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025347 
(FPCore (lamdp lamt ca one_es sa)
  :name "inverse-phi"
  :precision binary64
  (atan (/ (- (* (tan lamdp) (cos lamt)) (* ca (sin lamt))) (* one_es sa))))