VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.2%

Time: 19.5s

Alternatives: 31

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -6 \cdot 10^{+164}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + F \cdot F}} - t\_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -6e+164)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 50000.0)
       (/ (- (/ F (sqrt (+ 2.0 (* F F)))) t_0) (sin B))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -6e+164) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 50000.0) {
		tmp = ((F / sqrt((2.0 + (F * F)))) - t_0) / sin(B);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-6d+164)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= 50000.0d0) then
        tmp = ((f / sqrt((2.0d0 + (f * f)))) - t_0) / sin(b)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -6e+164) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= 50000.0) {
		tmp = ((F / Math.sqrt((2.0 + (F * F)))) - t_0) / Math.sin(B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -6e+164:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= 50000.0:
		tmp = ((F / math.sqrt((2.0 + (F * F)))) - t_0) / math.sin(B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -6e+164)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 50000.0)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(F * F)))) - t_0) / sin(B));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -6e+164)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= 50000.0)
		tmp = ((F / sqrt((2.0 + (F * F)))) - t_0) / sin(B);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e+164], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 50000.0], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.00000000000000001e164

   1. Initial program 20.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 33.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 33.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 33.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 33.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 33.7%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -6.00000000000000001e164 < F < 5e4

   1. Initial program 98.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   if 5e4 < F

   1. Initial program 62.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.2%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.3%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -68000000000000:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2}} - t\_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -68000000000000.0)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 0.00088)
       (/ (- (/ F (sqrt 2.0)) t_0) (sin B))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -68000000000000.0) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 0.00088) {
		tmp = ((F / sqrt(2.0)) - t_0) / sin(B);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-68000000000000.0d0)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= 0.00088d0) then
        tmp = ((f / sqrt(2.0d0)) - t_0) / sin(b)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -68000000000000.0) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= 0.00088) {
		tmp = ((F / Math.sqrt(2.0)) - t_0) / Math.sin(B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -68000000000000.0:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= 0.00088:
		tmp = ((F / math.sqrt(2.0)) - t_0) / math.sin(B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -68000000000000.0)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 0.00088)
		tmp = Float64(Float64(Float64(F / sqrt(2.0)) - t_0) / sin(B));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -68000000000000.0)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= 0.00088)
		tmp = ((F / sqrt(2.0)) - t_0) / sin(B);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -68000000000000.0], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], N[(N[(N[(F / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8e13

   1. Initial program 62.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 70.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 70.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 70.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 70.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 70.5%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -6.8e13 < F < 8.80000000000000031e-4

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(\sqrt{2}\right)}\right), \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]
   13. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{2}}} - x \cdot \cos B}{\sin B} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -0.021)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 0.00088)
       (- (* F (/ (sqrt 0.5) (sin B))) (/ x (tan B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 0.00088) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-0.021d0)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= 0.00088d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / tan(b))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= 0.00088) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -0.021:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= 0.00088:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / math.tan(B))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.021)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 0.00088)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.021)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= 0.00088)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / tan(B));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.021], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0210000000000000013

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -0.0210000000000000013 < F < 8.80000000000000031e-4

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{\sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sin-lowering-sin.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 99.3%

       \[\leadsto F \cdot \color{blue}{\frac{\sqrt{0.5}}{\sin B}} - \frac{x}{\tan B} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -0.021)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 7.2e-106)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* F F)))) (/ F B)) (/ x (tan B)))
       (if (<= F 0.00088)
         (- (/ (* F (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5)) (sin B)) (/ x B))
         (/ (- 1.0 t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 7.2e-106) {
		tmp = (sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - (x / tan(B));
	} else if (F <= 0.00088) {
		tmp = ((F * pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / sin(B)) - (x / B);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-0.021d0)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= 7.2d-106) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (f * f)))) * (f / b)) - (x / tan(b))
    else if (f <= 0.00088d0) then
        tmp = ((f * (((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0))) / sin(b)) - (x / b)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= 7.2e-106) {
		tmp = (Math.sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - (x / Math.tan(B));
	} else if (F <= 0.00088) {
		tmp = ((F * Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -0.021:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= 7.2e-106:
		tmp = (math.sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - (x / math.tan(B))
	elif F <= 0.00088:
		tmp = ((F * math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.021)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 7.2e-106)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 0.00088)
		tmp = Float64(Float64(Float64(F * (Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5)) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.021)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= 7.2e-106)
		tmp = (sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - (x / tan(B));
	elseif (F <= 0.00088)
		tmp = ((F * (((F * F) + (2.0 + (x * 2.0))) ^ -0.5)) / sin(B)) - (x / B);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.021], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e-106], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], N[(N[(N[(F * N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0210000000000000013

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -0.0210000000000000013 < F < 7.20000000000000025e-106

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. /-lowering-/.f64 87.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 87.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

   if 7.20000000000000025e-106 < F < 8.80000000000000031e-4

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 89.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   7. Simplified 89.9%

      \[\leadsto \frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \color{blue}{\frac{x}{B}} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{\sqrt{2 + F \cdot F}} - x}{\sin B}\\ t_1 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - t\_1}{\sin B}\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{t\_1}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (/ F (sqrt (+ 2.0 (* F F)))) x) (sin B)))
        (t_1 (* x (cos B))))
   (if (<= F -0.021)
     (/ (- -1.0 t_1) (sin B))
     (if (<= F -6.5e-169)
       t_0
       (if (<= F 3.4e-130)
         (/ t_1 (- 0.0 (sin B)))
         (if (<= F 0.00088) t_0 (/ (- 1.0 t_1) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sqrt((2.0 + (F * F)))) - x) / sin(B);
	double t_1 = x * cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_1) / sin(B);
	} else if (F <= -6.5e-169) {
		tmp = t_0;
	} else if (F <= 3.4e-130) {
		tmp = t_1 / (0.0 - sin(B));
	} else if (F <= 0.00088) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sqrt((2.0d0 + (f * f)))) - x) / sin(b)
    t_1 = x * cos(b)
    if (f <= (-0.021d0)) then
        tmp = ((-1.0d0) - t_1) / sin(b)
    else if (f <= (-6.5d-169)) then
        tmp = t_0
    else if (f <= 3.4d-130) then
        tmp = t_1 / (0.0d0 - sin(b))
    else if (f <= 0.00088d0) then
        tmp = t_0
    else
        tmp = (1.0d0 - t_1) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sqrt((2.0 + (F * F)))) - x) / Math.sin(B);
	double t_1 = x * Math.cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_1) / Math.sin(B);
	} else if (F <= -6.5e-169) {
		tmp = t_0;
	} else if (F <= 3.4e-130) {
		tmp = t_1 / (0.0 - Math.sin(B));
	} else if (F <= 0.00088) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sqrt((2.0 + (F * F)))) - x) / math.sin(B)
	t_1 = x * math.cos(B)
	tmp = 0
	if F <= -0.021:
		tmp = (-1.0 - t_1) / math.sin(B)
	elif F <= -6.5e-169:
		tmp = t_0
	elif F <= 3.4e-130:
		tmp = t_1 / (0.0 - math.sin(B))
	elif F <= 0.00088:
		tmp = t_0
	else:
		tmp = (1.0 - t_1) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(F * F)))) - x) / sin(B))
	t_1 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.021)
		tmp = Float64(Float64(-1.0 - t_1) / sin(B));
	elseif (F <= -6.5e-169)
		tmp = t_0;
	elseif (F <= 3.4e-130)
		tmp = Float64(t_1 / Float64(0.0 - sin(B)));
	elseif (F <= 0.00088)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sqrt((2.0 + (F * F)))) - x) / sin(B);
	t_1 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.021)
		tmp = (-1.0 - t_1) / sin(B);
	elseif (F <= -6.5e-169)
		tmp = t_0;
	elseif (F <= 3.4e-130)
		tmp = t_1 / (0.0 - sin(B));
	elseif (F <= 0.00088)
		tmp = t_0;
	else
		tmp = (1.0 - t_1) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.021], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.5e-169], t$95$0, If[LessEqual[F, 3.4e-130], N[(t$95$1 / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], t$95$0, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0210000000000000013

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -0.0210000000000000013 < F < -6.5000000000000002e-169 or 3.40000000000000005e-130 < F < 8.80000000000000031e-4

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 99.4%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in B around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \color{blue}{x}\right), \mathsf{sin.f64}\left(B\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 81.2%

       \[\leadsto \frac{\frac{F}{\sqrt{2 + F \cdot F}} - \color{blue}{x}}{\sin B} \]

   if -6.5000000000000002e-169 < F < 3.40000000000000005e-130

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \cos B\right)\right), \color{blue}{\sin B}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      6. sin-lowering-sin.f64 88.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x}{\sin B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{{\left(2 + F \cdot F\right)}^{0.5}}}{\sin B}\\ t_1 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.015:\\ \;\;\;\;\frac{-1 - t\_1}{\sin B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{t\_1}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00082:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (/ F (pow (+ 2.0 (* F F)) 0.5)) (sin B))) (t_1 (* x (cos B))))
   (if (<= F -0.015)
     (/ (- -1.0 t_1) (sin B))
     (if (<= F -1.5e-31)
       t_0
       (if (<= F 5.8e-78)
         (/ t_1 (- 0.0 (sin B)))
         (if (<= F 0.00082) t_0 (/ (- 1.0 t_1) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / pow((2.0 + (F * F)), 0.5)) / sin(B);
	double t_1 = x * cos(B);
	double tmp;
	if (F <= -0.015) {
		tmp = (-1.0 - t_1) / sin(B);
	} else if (F <= -1.5e-31) {
		tmp = t_0;
	} else if (F <= 5.8e-78) {
		tmp = t_1 / (0.0 - sin(B));
	} else if (F <= 0.00082) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (f / ((2.0d0 + (f * f)) ** 0.5d0)) / sin(b)
    t_1 = x * cos(b)
    if (f <= (-0.015d0)) then
        tmp = ((-1.0d0) - t_1) / sin(b)
    else if (f <= (-1.5d-31)) then
        tmp = t_0
    else if (f <= 5.8d-78) then
        tmp = t_1 / (0.0d0 - sin(b))
    else if (f <= 0.00082d0) then
        tmp = t_0
    else
        tmp = (1.0d0 - t_1) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F / Math.pow((2.0 + (F * F)), 0.5)) / Math.sin(B);
	double t_1 = x * Math.cos(B);
	double tmp;
	if (F <= -0.015) {
		tmp = (-1.0 - t_1) / Math.sin(B);
	} else if (F <= -1.5e-31) {
		tmp = t_0;
	} else if (F <= 5.8e-78) {
		tmp = t_1 / (0.0 - Math.sin(B));
	} else if (F <= 0.00082) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / math.pow((2.0 + (F * F)), 0.5)) / math.sin(B)
	t_1 = x * math.cos(B)
	tmp = 0
	if F <= -0.015:
		tmp = (-1.0 - t_1) / math.sin(B)
	elif F <= -1.5e-31:
		tmp = t_0
	elif F <= 5.8e-78:
		tmp = t_1 / (0.0 - math.sin(B))
	elif F <= 0.00082:
		tmp = t_0
	else:
		tmp = (1.0 - t_1) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / (Float64(2.0 + Float64(F * F)) ^ 0.5)) / sin(B))
	t_1 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.015)
		tmp = Float64(Float64(-1.0 - t_1) / sin(B));
	elseif (F <= -1.5e-31)
		tmp = t_0;
	elseif (F <= 5.8e-78)
		tmp = Float64(t_1 / Float64(0.0 - sin(B)));
	elseif (F <= 0.00082)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / ((2.0 + (F * F)) ^ 0.5)) / sin(B);
	t_1 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.015)
		tmp = (-1.0 - t_1) / sin(B);
	elseif (F <= -1.5e-31)
		tmp = t_0;
	elseif (F <= 5.8e-78)
		tmp = t_1 / (0.0 - sin(B));
	elseif (F <= 0.00082)
		tmp = t_0;
	else
		tmp = (1.0 - t_1) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[Power[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.015], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-31], t$95$0, If[LessEqual[F, 5.8e-78], N[(t$95$1 / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00082], t$95$0, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.014999999999999999

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -0.014999999999999999 < F < -1.49999999999999991e-31 or 5.8000000000000001e-78 < F < 8.1999999999999998e-4

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 77.9%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + F \cdot F}}\right), \color{blue}{\sin B}\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \frac{\sqrt{1}}{\sqrt{2 + F \cdot F}}\right), \sin B\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \frac{1}{\sqrt{2 + F \cdot F}}\right), \sin B\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{F}{\sqrt{2 + F \cdot F}}\right), \sin \color{blue}{B}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\sqrt{2 + F \cdot F}\right)\right), \sin \color{blue}{B}\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left({\left(2 + F \cdot F\right)}^{\frac{1}{2}}\right)\right), \sin B\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \mathsf{pow.f64}\left(\left(2 + F \cdot F\right), \frac{1}{2}\right)\right), \sin B\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \left(F \cdot F\right)\right), \frac{1}{2}\right)\right), \sin B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right), \frac{1}{2}\right)\right), \sin B\right) \]

      11. sin-lowering-sin.f64 78.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right), \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   7. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{{\left(2 + F \cdot F\right)}^{0.5}}}{\sin B}} \]

   if -1.49999999999999991e-31 < F < 5.8000000000000001e-78

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \cos B\right)\right), \color{blue}{\sin B}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      6. sin-lowering-sin.f64 74.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   if 8.1999999999999998e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.015:\\ \;\;\;\;\frac{-1 - x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{F}{{\left(2 + F \cdot F\right)}^{0.5}}}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00082:\\ \;\;\;\;\frac{\frac{F}{{\left(2 + F \cdot F\right)}^{0.5}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{{\left(2 + F \cdot F\right)}^{-0.5}}}\\ t_1 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - t\_1}{\sin B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{t\_1}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (/ (sin B) (pow (+ 2.0 (* F F)) -0.5)))) (t_1 (* x (cos B))))
   (if (<= F -0.021)
     (/ (- -1.0 t_1) (sin B))
     (if (<= F -2.1e-32)
       t_0
       (if (<= F 7.6e-75)
         (/ t_1 (- 0.0 (sin B)))
         (if (<= F 0.00088) t_0 (/ (- 1.0 t_1) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / (sin(B) / pow((2.0 + (F * F)), -0.5));
	double t_1 = x * cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_1) / sin(B);
	} else if (F <= -2.1e-32) {
		tmp = t_0;
	} else if (F <= 7.6e-75) {
		tmp = t_1 / (0.0 - sin(B));
	} else if (F <= 0.00088) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = f / (sin(b) / ((2.0d0 + (f * f)) ** (-0.5d0)))
    t_1 = x * cos(b)
    if (f <= (-0.021d0)) then
        tmp = ((-1.0d0) - t_1) / sin(b)
    else if (f <= (-2.1d-32)) then
        tmp = t_0
    else if (f <= 7.6d-75) then
        tmp = t_1 / (0.0d0 - sin(b))
    else if (f <= 0.00088d0) then
        tmp = t_0
    else
        tmp = (1.0d0 - t_1) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / (Math.sin(B) / Math.pow((2.0 + (F * F)), -0.5));
	double t_1 = x * Math.cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_1) / Math.sin(B);
	} else if (F <= -2.1e-32) {
		tmp = t_0;
	} else if (F <= 7.6e-75) {
		tmp = t_1 / (0.0 - Math.sin(B));
	} else if (F <= 0.00088) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / (math.sin(B) / math.pow((2.0 + (F * F)), -0.5))
	t_1 = x * math.cos(B)
	tmp = 0
	if F <= -0.021:
		tmp = (-1.0 - t_1) / math.sin(B)
	elif F <= -2.1e-32:
		tmp = t_0
	elif F <= 7.6e-75:
		tmp = t_1 / (0.0 - math.sin(B))
	elif F <= 0.00088:
		tmp = t_0
	else:
		tmp = (1.0 - t_1) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / Float64(sin(B) / (Float64(2.0 + Float64(F * F)) ^ -0.5)))
	t_1 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.021)
		tmp = Float64(Float64(-1.0 - t_1) / sin(B));
	elseif (F <= -2.1e-32)
		tmp = t_0;
	elseif (F <= 7.6e-75)
		tmp = Float64(t_1 / Float64(0.0 - sin(B)));
	elseif (F <= 0.00088)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / (sin(B) / ((2.0 + (F * F)) ^ -0.5));
	t_1 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.021)
		tmp = (-1.0 - t_1) / sin(B);
	elseif (F <= -2.1e-32)
		tmp = t_0;
	elseif (F <= 7.6e-75)
		tmp = t_1 / (0.0 - sin(B));
	elseif (F <= 0.00088)
		tmp = t_0;
	else
		tmp = (1.0 - t_1) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.021], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-32], t$95$0, If[LessEqual[F, 7.6e-75], N[(t$95$1 / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], t$95$0, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0210000000000000013

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -0.0210000000000000013 < F < -2.0999999999999999e-32 or 7.59999999999999987e-75 < F < 8.80000000000000031e-4

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 80.3%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto F \cdot \frac{1}{\color{blue}{\frac{\sin B}{\sqrt{\frac{1}{2 + F \cdot F}}}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{F}{\color{blue}{\frac{\sin B}{\sqrt{\frac{1}{2 + F \cdot F}}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \color{blue}{\left(\frac{\sin B}{\sqrt{\frac{1}{2 + F \cdot F}}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\sin B, \color{blue}{\left(\sqrt{\frac{1}{2 + F \cdot F}}\right)}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left(\sqrt{\color{blue}{\frac{1}{2 + F \cdot F}}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left(\sqrt{{\left(2 + F \cdot F\right)}^{-1}}\right)\right)\right) \]

      7. sqrt-pow1 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left({\left(2 + F \cdot F\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left({\left(2 + F \cdot F\right)}^{\frac{-1}{2}}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\left(2 + F \cdot F\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \left(F \cdot F\right)\right), \frac{-1}{2}\right)\right)\right) \]

      11. *-lowering-*.f64 80.2%

          \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right), \frac{-1}{2}\right)\right)\right) \]

   7. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{{\left(2 + F \cdot F\right)}^{-0.5}}}} \]

   if -2.0999999999999999e-32 < F < 7.59999999999999987e-75

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \cos B\right)\right), \color{blue}{\sin B}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      6. sin-lowering-sin.f64 74.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(2 + F \cdot F\right)}^{-0.5}}}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(2 + F \cdot F\right)}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -5.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-108}:\\ \;\;\;\;\frac{t\_0}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -5.1e-5)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F -2.9e-31)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 1.22e-108)
         (/ t_0 (- 0.0 (sin B)))
         (if (<= F 0.0007)
           (/
            (-
             (+
              (*
               (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
               (+ F (* 0.16666666666666666 (* F (* B B)))))
              (* B (* B (* x 0.3333333333333333))))
             x)
            B)
           (/ (- 1.0 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -5.1e-5) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= -2.9e-31) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 1.22e-108) {
		tmp = t_0 / (0.0 - sin(B));
	} else if (F <= 0.0007) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-5.1d-5)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= (-2.9d-31)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 1.22d-108) then
        tmp = t_0 / (0.0d0 - sin(b))
    else if (f <= 0.0007d0) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -5.1e-5) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= -2.9e-31) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 1.22e-108) {
		tmp = t_0 / (0.0 - Math.sin(B));
	} else if (F <= 0.0007) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -5.1e-5:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= -2.9e-31:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 1.22e-108:
		tmp = t_0 / (0.0 - math.sin(B))
	elif F <= 0.0007:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -5.1e-5)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= -2.9e-31)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 1.22e-108)
		tmp = Float64(t_0 / Float64(0.0 - sin(B)));
	elseif (F <= 0.0007)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -5.1e-5)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= -2.9e-31)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 1.22e-108)
		tmp = t_0 / (0.0 - sin(B));
	elseif (F <= 0.0007)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.1e-5], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.9e-31], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e-108], N[(t$95$0 / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0007], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.09999999999999996e-5

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -5.09999999999999996e-5 < F < -2.9000000000000001e-31

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]

   if -2.9000000000000001e-31 < F < 1.2199999999999999e-108

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \cos B\right)\right), \color{blue}{\sin B}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      6. sin-lowering-sin.f64 77.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   if 1.2199999999999999e-108 < F < 6.99999999999999993e-4

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}} \]

   if 6.99999999999999993e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-108}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.00023:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{t\_0}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00082:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -0.00023)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F -1.5e-31)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 1.1e-108)
         (/ t_0 (- 0.0 (sin B)))
         (if (<= F 0.00082)
           (/
            (-
             (+
              (*
               (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
               (+ F (* 0.16666666666666666 (* F (* B B)))))
              (* B (* B (* x 0.3333333333333333))))
             x)
            B)
           (- (/ 1.0 (sin B)) (/ x (tan B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -0.00023) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= -1.5e-31) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 1.1e-108) {
		tmp = t_0 / (0.0 - sin(B));
	} else if (F <= 0.00082) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-0.00023d0)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= (-1.5d-31)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 1.1d-108) then
        tmp = t_0 / (0.0d0 - sin(b))
    else if (f <= 0.00082d0) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -0.00023) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= -1.5e-31) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 1.1e-108) {
		tmp = t_0 / (0.0 - Math.sin(B));
	} else if (F <= 0.00082) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -0.00023:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= -1.5e-31:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 1.1e-108:
		tmp = t_0 / (0.0 - math.sin(B))
	elif F <= 0.00082:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.00023)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= -1.5e-31)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 1.1e-108)
		tmp = Float64(t_0 / Float64(0.0 - sin(B)));
	elseif (F <= 0.00082)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.00023)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= -1.5e-31)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 1.1e-108)
		tmp = t_0 / (0.0 - sin(B));
	elseif (F <= 0.00082)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00023], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-31], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-108], N[(t$95$0 / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00082], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.3000000000000001e-4

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -2.3000000000000001e-4 < F < -1.49999999999999991e-31

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]

   if -1.49999999999999991e-31 < F < 1.1000000000000001e-108

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \cos B\right)\right), \color{blue}{\sin B}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      6. sin-lowering-sin.f64 77.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   if 1.1000000000000001e-108 < F < 8.1999999999999998e-4

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}} \]

   if 8.1999999999999998e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00023:\\ \;\;\;\;\frac{-1 - x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00082:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0038:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-32}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0038)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.95e-32)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 1.55e-108)
         (/ (* x (cos B)) (- 0.0 (sin B)))
         (if (<= F 3.2e-6)
           (/
            (-
             (+
              (*
               (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
               (+ F (* 0.16666666666666666 (* F (* B B)))))
              (* B (* B (* x 0.3333333333333333))))
             x)
            B)
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0038) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.95e-32) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 1.55e-108) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 3.2e-6) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.0038d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.95d-32)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 1.55d-108) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 3.2d-6) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0038) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.95e-32) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 1.55e-108) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 3.2e-6) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0038:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.95e-32:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 1.55e-108:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 3.2e-6:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0038)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.95e-32)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 1.55e-108)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 3.2e-6)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0038)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.95e-32)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 1.55e-108)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 3.2e-6)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0038], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.95e-32], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e-108], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-6], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.00379999999999999999

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -0.00379999999999999999 < F < -1.9500000000000001e-32

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]

   if -1.9500000000000001e-32 < F < 1.55000000000000007e-108

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(x \cdot \cos B\right)\right), \color{blue}{\sin B}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

      6. sin-lowering-sin.f64 77.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

   if 1.55000000000000007e-108 < F < 3.1999999999999999e-6

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}} \]

   if 3.1999999999999999e-6 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0038:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-32}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.001:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{\cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.001)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.8e-32)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 5.4e-110)
         (* x (/ (cos B) (- 0.0 (sin B))))
         (if (<= F 5.8e-6)
           (/
            (-
             (+
              (*
               (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
               (+ F (* 0.16666666666666666 (* F (* B B)))))
              (* B (* B (* x 0.3333333333333333))))
             x)
            B)
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.001) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.8e-32) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 5.4e-110) {
		tmp = x * (cos(B) / (0.0 - sin(B)));
	} else if (F <= 5.8e-6) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.001d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.8d-32)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 5.4d-110) then
        tmp = x * (cos(b) / (0.0d0 - sin(b)))
    else if (f <= 5.8d-6) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.001) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.8e-32) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 5.4e-110) {
		tmp = x * (Math.cos(B) / (0.0 - Math.sin(B)));
	} else if (F <= 5.8e-6) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.001:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.8e-32:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 5.4e-110:
		tmp = x * (math.cos(B) / (0.0 - math.sin(B)))
	elif F <= 5.8e-6:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.001)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.8e-32)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 5.4e-110)
		tmp = Float64(x * Float64(cos(B) / Float64(0.0 - sin(B))));
	elseif (F <= 5.8e-6)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.001)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.8e-32)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 5.4e-110)
		tmp = x * (cos(B) / (0.0 - sin(B)));
	elseif (F <= 5.8e-6)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.001], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.8e-32], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e-110], N[(x * N[(N[Cos[B], $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1e-3

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -1e-3 < F < -1.79999999999999996e-32

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]

   if -1.79999999999999996e-32 < F < 5.3999999999999996e-110

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 99.6%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

   10. Taylor expanded in F around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{\left(\frac{\cos B}{\sin B}\right)}\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - x\right), \left(\frac{\color{blue}{\cos B}}{\sin B}\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{\color{blue}{\cos B}}{\sin B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\cos B, \color{blue}{\sin B}\right)\right) \]

       8. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \sin \color{blue}{B}\right)\right) \]

       9. sin-lowering-sin.f64 76.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   12. Simplified 76.8%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \frac{\cos B}{\sin B}} \]

   if 5.3999999999999996e-110 < F < 5.8000000000000004e-6

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}} \]

   if 5.8000000000000004e-6 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.001:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{\cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + F \cdot F\\ t_1 := x \cdot \cos B\\ \mathbf{if}\;F \leq -0.021:\\ \;\;\;\;\frac{-1 - t\_1}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\frac{1}{t\_0}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{\frac{F}{\sqrt{t\_0}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* F F))) (t_1 (* x (cos B))))
   (if (<= F -0.021)
     (/ (- -1.0 t_1) (sin B))
     (if (<= F 1.1e-105)
       (- (* (sqrt (/ 1.0 t_0)) (/ F B)) (/ x (tan B)))
       (if (<= F 0.00088)
         (/ (- (/ F (sqrt t_0)) x) (sin B))
         (/ (- 1.0 t_1) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (F * F);
	double t_1 = x * cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_1) / sin(B);
	} else if (F <= 1.1e-105) {
		tmp = (sqrt((1.0 / t_0)) * (F / B)) - (x / tan(B));
	} else if (F <= 0.00088) {
		tmp = ((F / sqrt(t_0)) - x) / sin(B);
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (f * f)
    t_1 = x * cos(b)
    if (f <= (-0.021d0)) then
        tmp = ((-1.0d0) - t_1) / sin(b)
    else if (f <= 1.1d-105) then
        tmp = (sqrt((1.0d0 / t_0)) * (f / b)) - (x / tan(b))
    else if (f <= 0.00088d0) then
        tmp = ((f / sqrt(t_0)) - x) / sin(b)
    else
        tmp = (1.0d0 - t_1) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (F * F);
	double t_1 = x * Math.cos(B);
	double tmp;
	if (F <= -0.021) {
		tmp = (-1.0 - t_1) / Math.sin(B);
	} else if (F <= 1.1e-105) {
		tmp = (Math.sqrt((1.0 / t_0)) * (F / B)) - (x / Math.tan(B));
	} else if (F <= 0.00088) {
		tmp = ((F / Math.sqrt(t_0)) - x) / Math.sin(B);
	} else {
		tmp = (1.0 - t_1) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (F * F)
	t_1 = x * math.cos(B)
	tmp = 0
	if F <= -0.021:
		tmp = (-1.0 - t_1) / math.sin(B)
	elif F <= 1.1e-105:
		tmp = (math.sqrt((1.0 / t_0)) * (F / B)) - (x / math.tan(B))
	elif F <= 0.00088:
		tmp = ((F / math.sqrt(t_0)) - x) / math.sin(B)
	else:
		tmp = (1.0 - t_1) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(F * F))
	t_1 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -0.021)
		tmp = Float64(Float64(-1.0 - t_1) / sin(B));
	elseif (F <= 1.1e-105)
		tmp = Float64(Float64(sqrt(Float64(1.0 / t_0)) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 0.00088)
		tmp = Float64(Float64(Float64(F / sqrt(t_0)) - x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (F * F);
	t_1 = x * cos(B);
	tmp = 0.0;
	if (F <= -0.021)
		tmp = (-1.0 - t_1) / sin(B);
	elseif (F <= 1.1e-105)
		tmp = (sqrt((1.0 / t_0)) * (F / B)) - (x / tan(B));
	elseif (F <= 0.00088)
		tmp = ((F / sqrt(t_0)) - x) / sin(B);
	else
		tmp = (1.0 - t_1) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.021], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-105], N[(N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], N[(N[(N[(F / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0210000000000000013

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 71.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 71.5%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(1 + x \cdot \cos B\right)}{\color{blue}{\sin B}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)\right), \color{blue}{\sin B}\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin \color{blue}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x \cdot \cos B\right)\right)\right), \sin B\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 - x \cdot \cos B\right), \sin \color{blue}{B}\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       9. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       10. sin-lowering-sin.f64 99.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   if -0.0210000000000000013 < F < 1.10000000000000002e-105

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. /-lowering-/.f64 87.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 87.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

   if 1.10000000000000002e-105 < F < 8.80000000000000031e-4

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 99.4%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in B around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \color{blue}{x}\right), \mathsf{sin.f64}\left(B\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 89.8%

       \[\leadsto \frac{\frac{F}{\sqrt{2 + F \cdot F}} - \color{blue}{x}}{\sin B} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 75.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 75.8%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

       2. associate-*l/ N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

       3. sub-div N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x \cdot \cos B}{\color{blue}{\sin B}} \]

       4. fmm-def N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(x \cdot \cos B\right)\right)}{\sin \color{blue}{B}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)}{\sin B} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + F \cdot F}}, \mathsf{neg}\left(\cos B \cdot x\right)\right)\right), \color{blue}{\sin B}\right) \]

   11. Applied egg-rr 75.9%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + F \cdot F}} - x \cdot \cos B}{\sin B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(1 - x \cdot \cos B\right), \color{blue}{\sin B}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \cos B\right)\right), \sin \color{blue}{B}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \cos B\right)\right), \sin B\right) \]

       4. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \sin B\right) \]

       5. sin-lowering-sin.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   14. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00088:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-32}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{\cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00088)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -4e-32)
     (/ (* F (sqrt 0.5)) (sin B))
     (if (<= F 4.6e-109)
       (* x (/ (cos B) (- 0.0 (sin B))))
       (if (<= F 0.00088)
         (/
          (-
           (+
            (*
             (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
             (+ F (* 0.16666666666666666 (* F (* B B)))))
            (* B (* B (* x 0.3333333333333333))))
           x)
          B)
         (- (* F (/ 1.0 (* F (sin B)))) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00088) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -4e-32) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 4.6e-109) {
		tmp = x * (cos(B) / (0.0 - sin(B)));
	} else if (F <= 0.00088) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.00088d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-4d-32)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 4.6d-109) then
        tmp = x * (cos(b) / (0.0d0 - sin(b)))
    else if (f <= 0.00088d0) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = (f * (1.0d0 / (f * sin(b)))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00088) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -4e-32) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 4.6e-109) {
		tmp = x * (Math.cos(B) / (0.0 - Math.sin(B)));
	} else if (F <= 0.00088) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * Math.sin(B)))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.00088:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -4e-32:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 4.6e-109:
		tmp = x * (math.cos(B) / (0.0 - math.sin(B)))
	elif F <= 0.00088:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = (F * (1.0 / (F * math.sin(B)))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00088)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -4e-32)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 4.6e-109)
		tmp = Float64(x * Float64(cos(B) / Float64(0.0 - sin(B))));
	elseif (F <= 0.00088)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.00088)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -4e-32)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 4.6e-109)
		tmp = x * (cos(B) / (0.0 - sin(B)));
	elseif (F <= 0.00088)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00088], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4e-32], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e-109], N[(x * N[(N[Cos[B], $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00088], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.80000000000000031e-4

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -8.80000000000000031e-4 < F < -4.00000000000000022e-32

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]

   if -4.00000000000000022e-32 < F < 4.6000000000000003e-109

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\frac{\sin B}{\color{blue}{\cos B}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \left(\frac{x}{\sin B} \cdot \color{blue}{\cos B}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{x}{\sin B}\right), \color{blue}{\cos B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \sin B\right), \cos \color{blue}{B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \cos B\right)\right) \]

      6. cos-lowering-cos.f64 99.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(B\right)\right), \mathsf{cos.f64}\left(B\right)\right)\right) \]

   9. Applied egg-rr 99.6%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

   10. Taylor expanded in F around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{\left(\frac{\cos B}{\sin B}\right)}\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - x\right), \left(\frac{\color{blue}{\cos B}}{\sin B}\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{\color{blue}{\cos B}}{\sin B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\cos B, \color{blue}{\sin B}\right)\right) \]

       8. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \sin \color{blue}{B}\right)\right) \]

       9. sin-lowering-sin.f64 76.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   12. Simplified 76.8%

       \[\leadsto \color{blue}{\left(0 - x\right) \cdot \frac{\cos B}{\sin B}} \]

   if 4.6000000000000003e-109 < F < 8.80000000000000031e-4

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}} \]

   if 8.80000000000000031e-4 < F

   1. Initial program 63.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 99.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 73.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   13. Simplified 73.5%

       \[\leadsto F \cdot \frac{1}{\sin B \cdot F} - \color{blue}{\frac{x}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00088:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-32}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{\cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00088:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00055:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \left(B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.00055)
   (/
    (-
     (+
      (*
       (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
       (+ F (* 0.16666666666666666 (* F (* B B)))))
      (* B (* B (* x 0.3333333333333333))))
     x)
    B)
   (-
    (*
     F
     (/
      1.0
      (*
       F
       (*
        B
        (+
         1.0
         (*
          (* B B)
          (+
           (*
            (* B B)
            (+ 0.008333333333333333 (* (* B B) -0.0001984126984126984)))
           -0.16666666666666666)))))))
    (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00055) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.00055d0) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = (f * (1.0d0 / (f * (b * (1.0d0 + ((b * b) * (((b * b) * (0.008333333333333333d0 + ((b * b) * (-0.0001984126984126984d0)))) + (-0.16666666666666666d0)))))))) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00055) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.00055:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.00055)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * Float64(B * Float64(1.0 + Float64(Float64(B * B) * Float64(Float64(Float64(B * B) * Float64(0.008333333333333333 + Float64(Float64(B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.00055)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.00055], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(B * B), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.50000000000000033e-4

   1. Initial program 79.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}} \]

   if 5.50000000000000033e-4 < B

   1. Initial program 88.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 88.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left(B \cdot \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)}, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {B}^{2}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({B}^{2} \cdot \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       16. *-lowering-*.f64 66.9%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 66.9%

       \[\leadsto F \cdot \frac{1}{\color{blue}{\left(B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)} \cdot F} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00055:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \left(B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00015:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \left(B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.00015)
   (/ (- (/ F (sqrt (+ 2.0 (+ (* F F) (* x 2.0))))) x) B)
   (-
    (*
     F
     (/
      1.0
      (*
       F
       (*
        B
        (+
         1.0
         (*
          (* B B)
          (+
           (*
            (* B B)
            (+ 0.008333333333333333 (* (* B B) -0.0001984126984126984)))
           -0.16666666666666666)))))))
    (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00015) {
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.00015d0) then
        tmp = ((f / sqrt((2.0d0 + ((f * f) + (x * 2.0d0))))) - x) / b
    else
        tmp = (f * (1.0d0 / (f * (b * (1.0d0 + ((b * b) * (((b * b) * (0.008333333333333333d0 + ((b * b) * (-0.0001984126984126984d0)))) + (-0.16666666666666666d0)))))))) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00015) {
		tmp = ((F / Math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.00015:
		tmp = ((F / math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B
	else:
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.00015)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * Float64(B * Float64(1.0 + Float64(Float64(B * B) * Float64(Float64(Float64(B * B) * Float64(0.008333333333333333 + Float64(Float64(B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.00015)
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	else
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * (((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984))) + -0.16666666666666666))))))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.00015], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(B * B), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.49999999999999987e-4

   1. Initial program 79.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   if 1.49999999999999987e-4 < B

   1. Initial program 88.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 88.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left(B \cdot \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)}, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {B}^{2}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({B}^{2} \cdot \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       16. *-lowering-*.f64 66.9%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 66.9%

       \[\leadsto F \cdot \frac{1}{\color{blue}{\left(B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)} \cdot F} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00015:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \left(B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-278}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -9.6e-177)
     t_0
     (if (<= x 6.5e-278)
       (* F (/ (sqrt (/ 1.0 (+ 2.0 (* F F)))) B))
       (if (<= x 1.7e-161)
         (/ 1.0 (sin B))
         (if (<= x 1.1e-24)
           (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
           t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -9.6e-177) {
		tmp = t_0;
	} else if (x <= 6.5e-278) {
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 1.7e-161) {
		tmp = 1.0 / sin(B);
	} else if (x <= 1.1e-24) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-9.6d-177)) then
        tmp = t_0
    else if (x <= 6.5d-278) then
        tmp = f * (sqrt((1.0d0 / (2.0d0 + (f * f)))) / b)
    else if (x <= 1.7d-161) then
        tmp = 1.0d0 / sin(b)
    else if (x <= 1.1d-24) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -9.6e-177) {
		tmp = t_0;
	} else if (x <= 6.5e-278) {
		tmp = F * (Math.sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 1.7e-161) {
		tmp = 1.0 / Math.sin(B);
	} else if (x <= 1.1e-24) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -9.6e-177:
		tmp = t_0
	elif x <= 6.5e-278:
		tmp = F * (math.sqrt((1.0 / (2.0 + (F * F)))) / B)
	elif x <= 1.7e-161:
		tmp = 1.0 / math.sin(B)
	elif x <= 1.1e-24:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -9.6e-177)
		tmp = t_0;
	elseif (x <= 6.5e-278)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) / B));
	elseif (x <= 1.7e-161)
		tmp = Float64(1.0 / sin(B));
	elseif (x <= 1.1e-24)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -9.6e-177)
		tmp = t_0;
	elseif (x <= 6.5e-278)
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	elseif (x <= 1.7e-161)
		tmp = 1.0 / sin(B);
	elseif (x <= 1.1e-24)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.6e-177], t$95$0, If[LessEqual[x, 6.5e-278], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7e-161], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-24], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.5999999999999995e-177 or 1.10000000000000001e-24 < x

   1. Initial program 81.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 90.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 90.3%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 67.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 67.8%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 76.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if -9.5999999999999995e-177 < x < 6.5e-278

   1. Initial program 79.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 68.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{B}}\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{B}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{B}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), B\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 47.6%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), B\right)\right) \]

   8. Simplified 47.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}} \]

   if 6.5e-278 < x < 1.69999999999999991e-161

   1. Initial program 80.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 69.9%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 44.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 1.69999999999999991e-161 < x < 1.10000000000000001e-24

   1. Initial program 82.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 50.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - \color{blue}{\frac{x}{B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), \left(\sqrt{\frac{1}{2 + 2 \cdot x}}\right)\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \left(\sqrt{\frac{1}{2 + 2 \cdot x}}\right)\right), \left(\frac{x}{B}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + 2 \cdot x}\right)\right)\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + 2 \cdot x\right)\right)\right)\right), \left(\frac{x}{B}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x\right)\right)\right)\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(x \cdot 2\right)\right)\right)\right)\right), \left(\frac{x}{B}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right)\right), \left(\frac{x}{B}\right)\right) \]

      12. /-lowering-/.f64 47.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-278}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -2.2e-176)
     t_0
     (if (<= x 2.45e-278)
       (* F (/ (sqrt (/ 1.0 (+ 2.0 (* F F)))) B))
       (if (<= x 3.6e-158)
         (/ 1.0 (sin B))
         (if (<= x 2e-25)
           (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
           t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -2.2e-176) {
		tmp = t_0;
	} else if (x <= 2.45e-278) {
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 3.6e-158) {
		tmp = 1.0 / sin(B);
	} else if (x <= 2e-25) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-2.2d-176)) then
        tmp = t_0
    else if (x <= 2.45d-278) then
        tmp = f * (sqrt((1.0d0 / (2.0d0 + (f * f)))) / b)
    else if (x <= 3.6d-158) then
        tmp = 1.0d0 / sin(b)
    else if (x <= 2d-25) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -2.2e-176) {
		tmp = t_0;
	} else if (x <= 2.45e-278) {
		tmp = F * (Math.sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 3.6e-158) {
		tmp = 1.0 / Math.sin(B);
	} else if (x <= 2e-25) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -2.2e-176:
		tmp = t_0
	elif x <= 2.45e-278:
		tmp = F * (math.sqrt((1.0 / (2.0 + (F * F)))) / B)
	elif x <= 3.6e-158:
		tmp = 1.0 / math.sin(B)
	elif x <= 2e-25:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -2.2e-176)
		tmp = t_0;
	elseif (x <= 2.45e-278)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) / B));
	elseif (x <= 3.6e-158)
		tmp = Float64(1.0 / sin(B));
	elseif (x <= 2e-25)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -2.2e-176)
		tmp = t_0;
	elseif (x <= 2.45e-278)
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	elseif (x <= 3.6e-158)
		tmp = 1.0 / sin(B);
	elseif (x <= 2e-25)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e-176], t$95$0, If[LessEqual[x, 2.45e-278], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-158], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-25], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.1999999999999999e-176 or 2.00000000000000008e-25 < x

   1. Initial program 81.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 90.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 90.3%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 67.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 67.8%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 76.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if -2.1999999999999999e-176 < x < 2.4500000000000001e-278

   1. Initial program 79.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 68.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{B}}\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{B}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{B}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), B\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 47.6%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), B\right)\right) \]

   8. Simplified 47.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}} \]

   if 2.4500000000000001e-278 < x < 3.59999999999999991e-158

   1. Initial program 80.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 69.9%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 44.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 3.59999999999999991e-158 < x < 2.00000000000000008e-25

   1. Initial program 82.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 50.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x\right)}, B\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\right), x\right), B\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + 2 \cdot x}}\right)\right), x\right), B\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + 2 \cdot x}\right)\right)\right), x\right), B\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + 2 \cdot x\right)\right)\right)\right), x\right), B\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(x \cdot 2\right)\right)\right)\right)\right), x\right), B\right) \]

      7. *-lowering-*.f64 47.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right)\right), x\right), B\right) \]

   8. Simplified 47.4%

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 57.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00012:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{B \cdot \left(F + \left(B \cdot B\right) \cdot \left(F \cdot \left(-0.16666666666666666 + \left(B \cdot B\right) \cdot 0.008333333333333333\right)\right)\right)} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.00012)
   (/ (- (/ F (sqrt (+ 2.0 (+ (* F F) (* x 2.0))))) x) B)
   (-
    (*
     F
     (/
      1.0
      (*
       B
       (+
        F
        (*
         (* B B)
         (* F (+ -0.16666666666666666 (* (* B B) 0.008333333333333333))))))))
    (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00012) {
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (F * (1.0 / (B * (F + ((B * B) * (F * (-0.16666666666666666 + ((B * B) * 0.008333333333333333)))))))) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.00012d0) then
        tmp = ((f / sqrt((2.0d0 + ((f * f) + (x * 2.0d0))))) - x) / b
    else
        tmp = (f * (1.0d0 / (b * (f + ((b * b) * (f * ((-0.16666666666666666d0) + ((b * b) * 0.008333333333333333d0)))))))) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00012) {
		tmp = ((F / Math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (F * (1.0 / (B * (F + ((B * B) * (F * (-0.16666666666666666 + ((B * B) * 0.008333333333333333)))))))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.00012:
		tmp = ((F / math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B
	else:
		tmp = (F * (1.0 / (B * (F + ((B * B) * (F * (-0.16666666666666666 + ((B * B) * 0.008333333333333333)))))))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.00012)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(B * Float64(F + Float64(Float64(B * B) * Float64(F * Float64(-0.16666666666666666 + Float64(Float64(B * B) * 0.008333333333333333)))))))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.00012)
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	else
		tmp = (F * (1.0 / (B * (F + ((B * B) * (F * (-0.16666666666666666 + ((B * B) * 0.008333333333333333)))))))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.00012], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(1.0 / N[(B * N[(F + N[(N[(B * B), $MachinePrecision] * N[(F * N[(-0.16666666666666666 + N[(N[(B * B), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.20000000000000003e-4

   1. Initial program 79.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   if 1.20000000000000003e-4 < B

   1. Initial program 88.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 88.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \color{blue}{\left(B \cdot \left(F + {B}^{2} \cdot \left(\frac{-1}{6} \cdot F + \frac{1}{120} \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \left(F + {B}^{2} \cdot \left(\frac{-1}{6} \cdot F + \frac{1}{120} \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \left({B}^{2} \cdot \left(\frac{-1}{6} \cdot F + \frac{1}{120} \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{-1}{6} \cdot F + \frac{1}{120} \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{-1}{6} \cdot F + \frac{1}{120} \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{-1}{6} \cdot F + \frac{1}{120} \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{120} \cdot \left({B}^{2} \cdot F\right) + \frac{-1}{6} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(\frac{1}{120} \cdot {B}^{2}\right) \cdot F + \frac{-1}{6} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       8. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(F \cdot \left(\frac{1}{120} \cdot {B}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(F \cdot \left(\frac{1}{120} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(F \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \left(\frac{1}{120} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \left(\frac{1}{120} \cdot {B}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(\frac{1}{120} \cdot {B}^{2}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left({B}^{2} \cdot \frac{1}{120}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \frac{1}{120}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \frac{1}{120}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       18. *-lowering-*.f64 66.9%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(F, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{1}{120}\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 66.9%

       \[\leadsto F \cdot \frac{1}{\color{blue}{B \cdot \left(F + \left(B \cdot B\right) \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot 0.008333333333333333 + -0.16666666666666666\right)\right)\right)}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00012:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{B \cdot \left(F + \left(B \cdot B\right) \cdot \left(F \cdot \left(-0.16666666666666666 + \left(B \cdot B\right) \cdot 0.008333333333333333\right)\right)\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-278}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;F \cdot \frac{\frac{1 + \frac{-1}{F \cdot F}}{F}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -9.5e-176)
     t_0
     (if (<= x 5.3e-278)
       (* F (/ (sqrt (/ 1.0 (+ 2.0 (* F F)))) B))
       (if (<= x 2.4e-27)
         (* F (/ (/ (+ 1.0 (/ -1.0 (* F F))) F) (sin B)))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -9.5e-176) {
		tmp = t_0;
	} else if (x <= 5.3e-278) {
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 2.4e-27) {
		tmp = F * (((1.0 + (-1.0 / (F * F))) / F) / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-9.5d-176)) then
        tmp = t_0
    else if (x <= 5.3d-278) then
        tmp = f * (sqrt((1.0d0 / (2.0d0 + (f * f)))) / b)
    else if (x <= 2.4d-27) then
        tmp = f * (((1.0d0 + ((-1.0d0) / (f * f))) / f) / sin(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -9.5e-176) {
		tmp = t_0;
	} else if (x <= 5.3e-278) {
		tmp = F * (Math.sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 2.4e-27) {
		tmp = F * (((1.0 + (-1.0 / (F * F))) / F) / Math.sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -9.5e-176:
		tmp = t_0
	elif x <= 5.3e-278:
		tmp = F * (math.sqrt((1.0 / (2.0 + (F * F)))) / B)
	elif x <= 2.4e-27:
		tmp = F * (((1.0 + (-1.0 / (F * F))) / F) / math.sin(B))
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -9.5e-176)
		tmp = t_0;
	elseif (x <= 5.3e-278)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) / B));
	elseif (x <= 2.4e-27)
		tmp = Float64(F * Float64(Float64(Float64(1.0 + Float64(-1.0 / Float64(F * F))) / F) / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -9.5e-176)
		tmp = t_0;
	elseif (x <= 5.3e-278)
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	elseif (x <= 2.4e-27)
		tmp = F * (((1.0 + (-1.0 / (F * F))) / F) / sin(B));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e-176], t$95$0, If[LessEqual[x, 5.3e-278], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-27], N[(F * N[(N[(N[(1.0 + N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5e-176 or 2.40000000000000002e-27 < x

   1. Initial program 81.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 90.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 67.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 67.4%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 75.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 75.7%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if -9.5e-176 < x < 5.3e-278

   1. Initial program 79.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 68.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{B}}\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{B}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{B}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), B\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 47.6%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), B\right)\right) \]

   8. Simplified 47.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}} \]

   if 5.3e-278 < x < 2.40000000000000002e-27

   1. Initial program 81.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 55.6%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 - \frac{1}{{F}^{2}}}{F}\right)}, \mathsf{sin.f64}\left(B\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - \frac{1}{{F}^{2}}\right), F\right), \mathsf{sin.f64}\left(\color{blue}{B}\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{{F}^{2}}\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{{F}^{2}}\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left({F}^{2}\right)\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(F \cdot F\right)\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(F, F\right)\right)\right), F\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   8. Simplified 36.6%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1 + \frac{-1}{F \cdot F}}{F}}}{\sin B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 57.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.000116:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \left(B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.000116)
   (/ (- (/ F (sqrt (+ 2.0 (+ (* F F) (* x 2.0))))) x) B)
   (-
    (* F (/ 1.0 (* F (* B (+ 1.0 (* (* B B) -0.16666666666666666))))))
    (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.000116) {
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * -0.16666666666666666)))))) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.000116d0) then
        tmp = ((f / sqrt((2.0d0 + ((f * f) + (x * 2.0d0))))) - x) / b
    else
        tmp = (f * (1.0d0 / (f * (b * (1.0d0 + ((b * b) * (-0.16666666666666666d0))))))) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.000116) {
		tmp = ((F / Math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * -0.16666666666666666)))))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.000116:
		tmp = ((F / math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B
	else:
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * -0.16666666666666666)))))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.000116)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * Float64(B * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666)))))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.000116)
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	else
		tmp = (F * (1.0 / (F * (B * (1.0 + ((B * B) * -0.16666666666666666)))))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.000116], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.16e-4

   1. Initial program 79.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   if 1.16e-4 < B

   1. Initial program 88.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 88.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       6. *-lowering-*.f64 66.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 66.3%

       \[\leadsto F \cdot \frac{1}{\color{blue}{\left(B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} \cdot F} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.000116:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \left(B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 57.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-278}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -5.1e-176)
     t_0
     (if (<= x 4.2e-278)
       (* F (/ (sqrt (/ 1.0 (+ 2.0 (* F F)))) B))
       (if (<= x 4e-35) (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -5.1e-176) {
		tmp = t_0;
	} else if (x <= 4.2e-278) {
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 4e-35) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-5.1d-176)) then
        tmp = t_0
    else if (x <= 4.2d-278) then
        tmp = f * (sqrt((1.0d0 / (2.0d0 + (f * f)))) / b)
    else if (x <= 4d-35) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -5.1e-176) {
		tmp = t_0;
	} else if (x <= 4.2e-278) {
		tmp = F * (Math.sqrt((1.0 / (2.0 + (F * F)))) / B);
	} else if (x <= 4e-35) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -5.1e-176:
		tmp = t_0
	elif x <= 4.2e-278:
		tmp = F * (math.sqrt((1.0 / (2.0 + (F * F)))) / B)
	elif x <= 4e-35:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -5.1e-176)
		tmp = t_0;
	elseif (x <= 4.2e-278)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) / B));
	elseif (x <= 4e-35)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -5.1e-176)
		tmp = t_0;
	elseif (x <= 4.2e-278)
		tmp = F * (sqrt((1.0 / (2.0 + (F * F)))) / B);
	elseif (x <= 4e-35)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.1e-176], t$95$0, If[LessEqual[x, 4.2e-278], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-35], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1000000000000003e-176 or 4.00000000000000003e-35 < x

   1. Initial program 81.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 90.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 67.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 67.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 75.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 75.2%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if -5.1000000000000003e-176 < x < 4.20000000000000027e-278

   1. Initial program 79.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 68.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{B}}\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{B}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{B}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), B\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 47.6%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), B\right)\right) \]

   8. Simplified 47.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + F \cdot F}}}{B}} \]

   if 4.20000000000000027e-278 < x < 4.00000000000000003e-35

   1. Initial program 81.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 56.6%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 56.6%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 37.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 56.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00022:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.00022)
   (/ (- (/ F (sqrt (+ 2.0 (+ (* F F) (* x 2.0))))) x) B)
   (- (/ 1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00022) {
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.00022d0) then
        tmp = ((f / sqrt((2.0d0 + ((f * f) + (x * 2.0d0))))) - x) / b
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.00022) {
		tmp = ((F / Math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.00022:
		tmp = ((F / math.sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.00022)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.00022)
		tmp = ((F / sqrt((2.0 + ((F * F) + (x * 2.0))))) - x) / B;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.00022], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.20000000000000008e-4

   1. Initial program 79.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   if 2.20000000000000008e-4 < B

   1. Initial program 88.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 88.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 62.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 62.2%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00022:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 43.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{elif}\;B \leq 0.000116:\\ \;\;\;\;\frac{B \cdot \frac{F}{F \cdot \left(1 + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F \cdot F}\right)} - x \cdot B}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 2.3e-165)
   (/ (- 0.0 x) B)
   (if (<= B 0.000116)
     (/
      (-
       (* B (/ F (* F (+ 1.0 (/ (* 0.5 (+ 2.0 (* x 2.0))) (* F F))))))
       (* x B))
      (* B B))
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.3e-165) {
		tmp = (0.0 - x) / B;
	} else if (B <= 0.000116) {
		tmp = ((B * (F / (F * (1.0 + ((0.5 * (2.0 + (x * 2.0))) / (F * F)))))) - (x * B)) / (B * B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 2.3d-165) then
        tmp = (0.0d0 - x) / b
    else if (b <= 0.000116d0) then
        tmp = ((b * (f / (f * (1.0d0 + ((0.5d0 * (2.0d0 + (x * 2.0d0))) / (f * f)))))) - (x * b)) / (b * b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.3e-165) {
		tmp = (0.0 - x) / B;
	} else if (B <= 0.000116) {
		tmp = ((B * (F / (F * (1.0 + ((0.5 * (2.0 + (x * 2.0))) / (F * F)))))) - (x * B)) / (B * B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 2.3e-165:
		tmp = (0.0 - x) / B
	elif B <= 0.000116:
		tmp = ((B * (F / (F * (1.0 + ((0.5 * (2.0 + (x * 2.0))) / (F * F)))))) - (x * B)) / (B * B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 2.3e-165)
		tmp = Float64(Float64(0.0 - x) / B);
	elseif (B <= 0.000116)
		tmp = Float64(Float64(Float64(B * Float64(F / Float64(F * Float64(1.0 + Float64(Float64(0.5 * Float64(2.0 + Float64(x * 2.0))) / Float64(F * F)))))) - Float64(x * B)) / Float64(B * B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 2.3e-165)
		tmp = (0.0 - x) / B;
	elseif (B <= 0.000116)
		tmp = ((B * (F / (F * (1.0 + ((0.5 * (2.0 + (x * 2.0))) / (F * F)))))) - (x * B)) / (B * B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 2.3e-165], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 0.000116], N[(N[(N[(B * N[(F / N[(F * N[(1.0 + N[(N[(0.5 * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * B), $MachinePrecision]), $MachinePrecision] / N[(B * B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.3e-165

   1. Initial program 79.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 30.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 30.8%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 2.3e-165 < B < 1.16e-4

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}}{B} - \color{blue}{\frac{x}{B}} \]

      2. frac-sub N/A

         \[\leadsto \frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}\right) \cdot B - B \cdot x}{\color{blue}{B \cdot B}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}\right) \cdot B - B \cdot x\right), \color{blue}{\left(B \cdot B\right)}\right) \]

   7. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot B - B \cdot x}{B \cdot B}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)}\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

      10. *-lowering-*.f64 71.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), B\right), \mathsf{*.f64}\left(B, x\right)\right), \mathsf{*.f64}\left(B, B\right)\right) \]

   10. Simplified 71.3%

       \[\leadsto \frac{\frac{F}{\color{blue}{F \cdot \left(1 + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F \cdot F}\right)}} \cdot B - B \cdot x}{B \cdot B} \]

   if 1.16e-4 < B

   1. Initial program 88.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 88.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 62.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 62.2%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{elif}\;B \leq 0.000116:\\ \;\;\;\;\frac{B \cdot \frac{F}{F \cdot \left(1 + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F \cdot F}\right)} - x \cdot B}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 44.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.22 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.22e+21)
   (/ -1.0 (sin B))
   (if (<= F 3.7e-14) (/ (- 0.0 x) B) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.22e+21) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3.7e-14) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.22d+21)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 3.7d-14) then
        tmp = (0.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.22e+21) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 3.7e-14) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.22e+21:
		tmp = -1.0 / math.sin(B)
	elif F <= 3.7e-14:
		tmp = (0.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.22e+21)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3.7e-14)
		tmp = Float64(Float64(0.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.22e+21)
		tmp = -1.0 / sin(B);
	elseif (F <= 3.7e-14)
		tmp = (0.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.22e+21], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.7e-14], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.22e21

   1. Initial program 59.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 27.1%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 27.1%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 57.9%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.22e21 < F < 3.70000000000000001e-14

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 52.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 33.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 3.70000000000000001e-14 < F

   1. Initial program 66.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 36.2%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 36.2%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 52.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 42.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.22 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(B \cdot B\right) \cdot \left(\left(0.019444444444444445 + x \cdot 0.022222222222222223\right) + \left(B \cdot B\right) \cdot \left(\left(x \cdot 0.022222222222222223\right) \cdot -0.3333333333333333 + \left(x \cdot 0.009523809523809525 + 0.00205026455026455\right)\right)\right)\right) - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.22e+21)
   (/ -1.0 (sin B))
   (if (<= F 1.2e-77)
     (/ (- 0.0 x) B)
     (/
      (+
       1.0
       (-
        (*
         (* B B)
         (+
          (+ 0.16666666666666666 (* x 0.3333333333333333))
          (*
           (* B B)
           (+
            (+ 0.019444444444444445 (* x 0.022222222222222223))
            (*
             (* B B)
             (+
              (* (* x 0.022222222222222223) -0.3333333333333333)
              (+ (* x 0.009523809523809525) 0.00205026455026455)))))))
        x))
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.22e+21) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.2e-77) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.22d+21)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.2d-77) then
        tmp = (0.0d0 - x) / b
    else
        tmp = (1.0d0 + (((b * b) * ((0.16666666666666666d0 + (x * 0.3333333333333333d0)) + ((b * b) * ((0.019444444444444445d0 + (x * 0.022222222222222223d0)) + ((b * b) * (((x * 0.022222222222222223d0) * (-0.3333333333333333d0)) + ((x * 0.009523809523809525d0) + 0.00205026455026455d0))))))) - x)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.22e+21) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.2e-77) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.22e+21:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.2e-77:
		tmp = (0.0 - x) / B
	else:
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.22e+21)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.2e-77)
		tmp = Float64(Float64(0.0 - x) / B);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(B * B) * Float64(Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)) + Float64(Float64(B * B) * Float64(Float64(0.019444444444444445 + Float64(x * 0.022222222222222223)) + Float64(Float64(B * B) * Float64(Float64(Float64(x * 0.022222222222222223) * -0.3333333333333333) + Float64(Float64(x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.22e+21)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.2e-77)
		tmp = (0.0 - x) / B;
	else
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.22e+21], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-77], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(B * B), $MachinePrecision] * N[(N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(0.019444444444444445 + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(N[(x * 0.022222222222222223), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[(x * 0.009523809523809525), $MachinePrecision] + 0.00205026455026455), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.22e21

   1. Initial program 59.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 27.1%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 27.1%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 57.9%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.22e21 < F < 1.19999999999999995e-77

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 52.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 52.0%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 35.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 1.19999999999999995e-77 < F

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 80.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 80.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 85.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 85.9%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{31}{15120} + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   13. Simplified 40.8%

       \[\leadsto \color{blue}{\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(B \cdot B\right) \cdot \left(\left(0.019444444444444445 + x \cdot 0.022222222222222223\right) + \left(B \cdot B\right) \cdot \left(\left(x \cdot 0.022222222222222223\right) \cdot -0.3333333333333333 + \left(x \cdot 0.009523809523809525 + 0.00205026455026455\right)\right)\right)\right) - x\right)}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 43.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(B \cdot B\right) \cdot \left(\left(0.019444444444444445 + x \cdot 0.022222222222222223\right) + \left(B \cdot B\right) \cdot \left(\left(x \cdot 0.022222222222222223\right) \cdot -0.3333333333333333 + \left(x \cdot 0.009523809523809525 + 0.00205026455026455\right)\right)\right)\right) - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-77)
   (/ (- -1.0 x) B)
   (if (<= F 1.2e-77)
     (/ (- 0.0 x) B)
     (/
      (+
       1.0
       (-
        (*
         (* B B)
         (+
          (+ 0.16666666666666666 (* x 0.3333333333333333))
          (*
           (* B B)
           (+
            (+ 0.019444444444444445 (* x 0.022222222222222223))
            (*
             (* B B)
             (+
              (* (* x 0.022222222222222223) -0.3333333333333333)
              (+ (* x 0.009523809523809525) 0.00205026455026455)))))))
        x))
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-77) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e-77) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.4d-77)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.2d-77) then
        tmp = (0.0d0 - x) / b
    else
        tmp = (1.0d0 + (((b * b) * ((0.16666666666666666d0 + (x * 0.3333333333333333d0)) + ((b * b) * ((0.019444444444444445d0 + (x * 0.022222222222222223d0)) + ((b * b) * (((x * 0.022222222222222223d0) * (-0.3333333333333333d0)) + ((x * 0.009523809523809525d0) + 0.00205026455026455d0))))))) - x)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-77) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e-77) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-77:
		tmp = (-1.0 - x) / B
	elif F <= 1.2e-77:
		tmp = (0.0 - x) / B
	else:
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-77)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.2e-77)
		tmp = Float64(Float64(0.0 - x) / B);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(B * B) * Float64(Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)) + Float64(Float64(B * B) * Float64(Float64(0.019444444444444445 + Float64(x * 0.022222222222222223)) + Float64(Float64(B * B) * Float64(Float64(Float64(x * 0.022222222222222223) * -0.3333333333333333) + Float64(Float64(x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e-77)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.2e-77)
		tmp = (0.0 - x) / B;
	else
		tmp = (1.0 + (((B * B) * ((0.16666666666666666 + (x * 0.3333333333333333)) + ((B * B) * ((0.019444444444444445 + (x * 0.022222222222222223)) + ((B * B) * (((x * 0.022222222222222223) * -0.3333333333333333) + ((x * 0.009523809523809525) + 0.00205026455026455))))))) - x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-77], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.2e-77], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(B * B), $MachinePrecision] * N[(N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(0.019444444444444445 + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] + N[(N[(B * B), $MachinePrecision] * N[(N[(N[(x * 0.022222222222222223), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[(x * 0.009523809523809525), $MachinePrecision] + 0.00205026455026455), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3999999999999999e-77

   1. Initial program 71.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 39.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 42.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -2.3999999999999999e-77 < F < 1.19999999999999995e-77

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 50.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 38.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 1.19999999999999995e-77 < F

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. sin-lowering-sin.f64 80.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 80.5%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 85.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 85.9%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{31}{15120} + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   13. Simplified 40.8%

       \[\leadsto \color{blue}{\frac{1 + \left(\left(B \cdot B\right) \cdot \left(\left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(B \cdot B\right) \cdot \left(\left(0.019444444444444445 + x \cdot 0.022222222222222223\right) + \left(B \cdot B\right) \cdot \left(\left(x \cdot 0.022222222222222223\right) \cdot -0.3333333333333333 + \left(x \cdot 0.009523809523809525 + 0.00205026455026455\right)\right)\right)\right) - x\right)}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 36.7% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - x}{B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 x) B)))
   (if (<= F -6.2e-74)
     (/ (- -1.0 x) B)
     (if (<= F 1.15e-14) t_0 (if (<= F 3.4e+232) (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (0.0 - x) / B;
	double tmp;
	if (F <= -6.2e-74) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e-14) {
		tmp = t_0;
	} else if (F <= 3.4e+232) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.0d0 - x) / b
    if (f <= (-6.2d-74)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.15d-14) then
        tmp = t_0
    else if (f <= 3.4d+232) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (0.0 - x) / B;
	double tmp;
	if (F <= -6.2e-74) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e-14) {
		tmp = t_0;
	} else if (F <= 3.4e+232) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (0.0 - x) / B
	tmp = 0
	if F <= -6.2e-74:
		tmp = (-1.0 - x) / B
	elif F <= 1.15e-14:
		tmp = t_0
	elif F <= 3.4e+232:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(0.0 - x) / B)
	tmp = 0.0
	if (F <= -6.2e-74)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.15e-14)
		tmp = t_0;
	elseif (F <= 3.4e+232)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (0.0 - x) / B;
	tmp = 0.0;
	if (F <= -6.2e-74)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.15e-14)
		tmp = t_0;
	elseif (F <= 3.4e+232)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -6.2e-74], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.15e-14], t$95$0, If[LessEqual[F, 3.4e+232], N[(1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2000000000000003e-74

   1. Initial program 71.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 39.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 42.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -6.2000000000000003e-74 < F < 1.14999999999999999e-14 or 3.3999999999999998e232 < F

   1. Initial program 85.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 48.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 36.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 1.14999999999999999e-14 < F < 3.3999999999999998e232

   1. Initial program 82.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 51.9%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \color{blue}{\left(F \cdot \sin B\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot \color{blue}{F}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, \color{blue}{F}\right)\right)\right) \]

      4. sin-lowering-sin.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right) \]

   8. Simplified 59.0%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1}{B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 31.5%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   11. Simplified 31.5%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 43.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e-74)
   (/ (- -1.0 x) B)
   (if (<= F 5.8e-78) (/ (- 0.0 x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e-74) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.8e-78) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.7d-74)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.8d-78) then
        tmp = (0.0d0 - x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e-74) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.8e-78) {
		tmp = (0.0 - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.7e-74:
		tmp = (-1.0 - x) / B
	elif F <= 5.8e-78:
		tmp = (0.0 - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e-74)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.8e-78)
		tmp = Float64(Float64(0.0 - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.7e-74)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.8e-78)
		tmp = (0.0 - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e-74], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.8e-78], N[(N[(0.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.70000000000000018e-74

   1. Initial program 71.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 39.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 42.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -2.70000000000000018e-74 < F < 5.8000000000000001e-78

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 50.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{B}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 38.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 5.8000000000000001e-78 < F

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 38.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 40.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   8. Simplified 40.7%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 30.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+235}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F 3.5e-14) t_0 (if (<= F 4.1e+235) (/ 1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= 3.5e-14) {
		tmp = t_0;
	} else if (F <= 4.1e+235) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= 3.5d-14) then
        tmp = t_0
    else if (f <= 4.1d+235) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= 3.5e-14) {
		tmp = t_0;
	} else if (F <= 4.1e+235) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= 3.5e-14:
		tmp = t_0
	elif F <= 4.1e+235:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= 3.5e-14)
		tmp = t_0;
	elseif (F <= 4.1e+235)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= 3.5e-14)
		tmp = t_0;
	elseif (F <= 4.1e+235)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, 3.5e-14], t$95$0, If[LessEqual[F, 4.1e+235], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if F < 3.5000000000000002e-14 or 4.1000000000000002e235 < F

   1. Initial program 81.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 45.4%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 28.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 28.2%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if 3.5000000000000002e-14 < F < 4.1000000000000002e235

   1. Initial program 82.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 51.9%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \color{blue}{\left(F \cdot \sin B\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot \color{blue}{F}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, \color{blue}{F}\right)\right)\right) \]

      4. sin-lowering-sin.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right) \]

   8. Simplified 59.0%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1}{B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 31.5%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   11. Simplified 31.5%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 17.2% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 6.2e-136) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.2e-136) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 6.2d-136) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.2e-136) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 6.2e-136:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 6.2e-136)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 6.2e-136)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 6.2e-136], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.2e-136

   1. Initial program 86.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 45.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{\color{blue}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 28.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 13.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{B}\right) \]

   11. Simplified 13.0%

       \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if 6.2e-136 < F

   1. Initial program 74.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\color{blue}{\sin B}} \]

      2. associate-/l* N/A

         \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 40.1%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 40.1%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \color{blue}{\left(F \cdot \sin B\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot \color{blue}{F}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, \color{blue}{F}\right)\right)\right) \]

      4. sin-lowering-sin.f64 40.8%

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right) \]

   8. Simplified 40.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1}{B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 21.3%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   11. Simplified 21.3%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 10.3% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 81.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   11. *-lowering-*.f64 43.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

5. Simplified 43.3%

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

6. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{\color{blue}{B}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

   7. --lowering--.f64 23.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

8. Simplified 23.7%

   \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 8.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{B}\right) \]

11. Simplified 8.6%

    \[\leadsto \color{blue}{\frac{-1}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))