VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.5%

Time: 19.6s

Alternatives: 35

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: 76.7% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3e+60)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.14)
       (- (/ F (* (sin B) (sqrt (+ (* F F) (+ 2.0 (* x 2.0)))))) t_0)
       (- (/ 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 <= -3e+60) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = (F / (sin(B) * sqrt(((F * F) + (2.0 + (x * 2.0)))))) - t_0;
	} 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 <= (-3d+60)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.14d0) then
        tmp = (f / (sin(b) * sqrt(((f * f) + (2.0d0 + (x * 2.0d0)))))) - t_0
    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 <= -3e+60) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = (F / (Math.sin(B) * Math.sqrt(((F * F) + (2.0 + (x * 2.0)))))) - t_0;
	} 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 <= -3e+60:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.14:
		tmp = (F / (math.sin(B) * math.sqrt(((F * F) + (2.0 + (x * 2.0)))))) - t_0
	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 <= -3e+60)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.14)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	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 <= -3e+60)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.14)
		tmp = (F / (sin(B) * sqrt(((F * F) + (2.0 + (x * 2.0)))))) - t_0;
	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, -3e+60], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.9999999999999998e60

   1. Initial program 60.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 74.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 99.7%

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

   9. Simplified 99.7%

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

   if -2.9999999999999998e60 < F < 0.14000000000000001

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   if 0.14000000000000001 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;\frac{{\left(F \cdot F + 2\right)}^{-0.5}}{\frac{\sin B}{F}} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+31)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.2)
       (+ (/ (pow (+ (* F F) 2.0) -0.5) (/ (sin B) F)) (/ -1.0 (/ (tan B) x)))
       (- (/ 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 <= -1e+31) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.2) {
		tmp = (pow(((F * F) + 2.0), -0.5) / (sin(B) / F)) + (-1.0 / (tan(B) / x));
	} 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 <= (-1d+31)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.2d0) then
        tmp = ((((f * f) + 2.0d0) ** (-0.5d0)) / (sin(b) / f)) + ((-1.0d0) / (tan(b) / x))
    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 <= -1e+31) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.2) {
		tmp = (Math.pow(((F * F) + 2.0), -0.5) / (Math.sin(B) / F)) + (-1.0 / (Math.tan(B) / x));
	} 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 <= -1e+31:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.2:
		tmp = (math.pow(((F * F) + 2.0), -0.5) / (math.sin(B) / F)) + (-1.0 / (math.tan(B) / x))
	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 <= -1e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.2)
		tmp = Float64(Float64((Float64(Float64(F * F) + 2.0) ^ -0.5) / Float64(sin(B) / F)) + Float64(-1.0 / Float64(tan(B) / x)));
	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 <= -1e+31)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.2)
		tmp = ((((F * F) + 2.0) ^ -0.5) / (sin(B) / F)) + (-1.0 / (tan(B) / x));
	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, -1e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.2], N[(N[(N[Power[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.9999999999999996e30

   1. Initial program 63.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 76.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 99.7%

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

   9. Simplified 99.7%

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

   if -9.9999999999999996e30 < F < 0.20000000000000001

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

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\frac{\sin B}{F}} + \frac{-1}{\frac{\tan B}{x}}} \]

   5. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \frac{{\color{blue}{\left(2 + F \cdot F\right)}}^{-0.5}}{\frac{\sin B}{F}} + \frac{-1}{\frac{\tan B}{x}} \]

   if 0.20000000000000001 < F

   1. Initial program 52.1%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;\frac{{\left(F \cdot F + 2\right)}^{-0.5}}{\frac{\sin B}{F}} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.92:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)\right)} - t\_0\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.92)
     (-
      (/ F (* (sin B) (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))))
      t_0)
     (if (<= F 0.14)
       (- (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 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 <= -0.92) {
		tmp = (F / (sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0;
	} else if (F <= 0.14) {
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	} 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 <= (-0.92d0)) then
        tmp = (f / (sin(b) * (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f)))))) - t_0
    else if (f <= 0.14d0) then
        tmp = (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    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 <= -0.92) {
		tmp = (F / (Math.sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0;
	} else if (F <= 0.14) {
		tmp = (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} 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 <= -0.92:
		tmp = (F / (math.sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0
	elif F <= 0.14:
		tmp = (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0))))) - t_0
	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 <= -0.92)
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F)))))) - t_0);
	elseif (F <= 0.14)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	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 <= -0.92)
		tmp = (F / (sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0;
	elseif (F <= 0.14)
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	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, -0.92], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F * N[(-1.0 + N[(N[(-1.0 - N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.92000000000000004

   1. Initial program 66.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 78.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 78.5%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \left(0 - F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      15. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 99.2%

      \[\leadsto \frac{F}{\sin B \cdot \color{blue}{\left(0 - F \cdot \left(1 + \frac{1 + 0.5 \cdot \left(x \cdot 2\right)}{F \cdot F}\right)\right)}} - \frac{x}{\tan B} \]

   if -0.92000000000000004 < F < 0.14000000000000001

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 99.4%

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

   9. Simplified 99.4%

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

   if 0.14000000000000001 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.92:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-151}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{t\_0}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00027:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{t\_0}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))) (t_1 (/ x (tan B))))
   (if (<= F -0.00049)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4e-151)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 t_0))) (/ x B))
       (if (<= F 3.5e-146)
         (/ (* x (cos B)) (- 0.0 (sin B)))
         (if (<= F 0.00027)
           (- (/ F (* (sin B) (sqrt t_0))) (/ x B))
           (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.00049) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4e-151) {
		tmp = ((F / sin(B)) * sqrt((1.0 / t_0))) - (x / B);
	} else if (F <= 3.5e-146) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 0.00027) {
		tmp = (F / (sin(B) * sqrt(t_0))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	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 + (x * 2.0d0)
    t_1 = x / tan(b)
    if (f <= (-0.00049d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4d-151)) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / t_0))) - (x / b)
    else if (f <= 3.5d-146) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 0.00027d0) then
        tmp = (f / (sin(b) * sqrt(t_0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00049) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4e-151) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / t_0))) - (x / B);
	} else if (F <= 3.5e-146) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 0.00027) {
		tmp = (F / (Math.sin(B) * Math.sqrt(t_0))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.00049:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4e-151:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / t_0))) - (x / B)
	elif F <= 3.5e-146:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 0.00027:
		tmp = (F / (math.sin(B) * math.sqrt(t_0))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00049)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4e-151)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / t_0))) - Float64(x / B));
	elseif (F <= 3.5e-146)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 0.00027)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(t_0))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00049)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4e-151)
		tmp = ((F / sin(B)) * sqrt((1.0 / t_0))) - (x / B);
	elseif (F <= 3.5e-146)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 0.00027)
		tmp = (F / (sin(B) * sqrt(t_0))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00049], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4e-151], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-146], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00027], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.8999999999999998e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -4.8999999999999998e-4 < F < -3.9999999999999998e-151

   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

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

      1. /-lowering-/.f64 79.9%

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

   5. Simplified 79.9%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 79.4%

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

   8. Simplified 79.4%

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

   if -3.9999999999999998e-151 < F < 3.5000000000000001e-146

   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. 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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(0 - \color{blue}{\sin B}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\sin B}\right)\right) \]

      8. sin-lowering-sin.f64 88.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{0 - \sin B}} \]

   if 3.5000000000000001e-146 < F < 2.70000000000000003e-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. Add Preprocessing

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 76.4%

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

   5. Simplified 76.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

   7. Applied egg-rr 76.7%

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

   8. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 76.2%

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

   10. Simplified 76.2%

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

   if 2.70000000000000003e-4 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-151}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00027:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00022:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-143}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.000195:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -0.00022)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4.5e-151)
       t_0
       (if (<= F 1.02e-143)
         (/ (* x (cos B)) (- 0.0 (sin B)))
         (if (<= F 0.000195) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.00022) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4.5e-151) {
		tmp = t_0;
	} else if (F <= 1.02e-143) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 0.000195) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	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) * sqrt((2.0d0 + (x * 2.0d0))))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-0.00022d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4.5d-151)) then
        tmp = t_0
    else if (f <= 1.02d-143) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 0.000195d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    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.sqrt((2.0 + (x * 2.0))))) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00022) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4.5e-151) {
		tmp = t_0;
	} else if (F <= 1.02e-143) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 0.000195) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0))))) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.00022:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4.5e-151:
		tmp = t_0
	elif F <= 1.02e-143:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 0.000195:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00022)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4.5e-151)
		tmp = t_0;
	elseif (F <= 1.02e-143)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 0.000195)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00022)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4.5e-151)
		tmp = t_0;
	elseif (F <= 1.02e-143)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 0.000195)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00022], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4.5e-151], t$95$0, If[LessEqual[F, 1.02e-143], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.000195], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.20000000000000008e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -2.20000000000000008e-4 < F < -4.5000000000000002e-151 or 1.02e-143 < F < 1.94999999999999996e-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. Add Preprocessing

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 78.3%

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

   5. Simplified 78.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

   7. Applied egg-rr 78.3%

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

   8. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 77.9%

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

   10. Simplified 77.9%

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

   if -4.5000000000000002e-151 < F < 1.02e-143

   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. 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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(0 - \color{blue}{\sin B}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\sin B}\right)\right) \]

      8. sin-lowering-sin.f64 88.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{0 - \sin B}} \]

   if 1.94999999999999996e-4 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

Alternative 6: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00042:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)\right)} - t\_0\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-151}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00042)
     (-
      (/ F (* (sin B) (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))))
      t_0)
     (if (<= F -4.8e-151)
       (- (* F (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B))) (/ x B))
       (if (<= F 0.14)
         (- (/ F (* B (sqrt (+ (* x 2.0) (+ (* F F) 2.0))))) t_0)
         (- (/ 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 <= -0.00042) {
		tmp = (F / (sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0;
	} else if (F <= -4.8e-151) {
		tmp = (F * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= (-0.00042d0)) then
        tmp = (f / (sin(b) * (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f)))))) - t_0
    else if (f <= (-4.8d-151)) then
        tmp = (f * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = (f / (b * sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))))) - t_0
    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 <= -0.00042) {
		tmp = (F / (Math.sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0;
	} else if (F <= -4.8e-151) {
		tmp = (F * (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * Math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= -0.00042:
		tmp = (F / (math.sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0
	elif F <= -4.8e-151:
		tmp = (F * (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - (x / B)
	elif F <= 0.14:
		tmp = (F / (B * math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0
	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 <= -0.00042)
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F)))))) - t_0);
	elseif (F <= -4.8e-151)
		tmp = Float64(Float64(F * Float64((Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5) / sin(B))) - Float64(x / B));
	elseif (F <= 0.14)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))))) - t_0);
	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 <= -0.00042)
		tmp = (F / (sin(B) * (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F)))))) - t_0;
	elseif (F <= -4.8e-151)
		tmp = (F * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - (x / B);
	elseif (F <= 0.14)
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	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, -0.00042], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F * N[(-1.0 + N[(N[(-1.0 - N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.8e-151], N[(N[(F * N[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F / N[(B * N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.2000000000000002e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \left(0 - F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      15. *-lowering-*.f64 99.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 99.2%

      \[\leadsto \frac{F}{\sin B \cdot \color{blue}{\left(0 - F \cdot \left(1 + \frac{1 + 0.5 \cdot \left(x \cdot 2\right)}{F \cdot F}\right)\right)}} - \frac{x}{\tan B} \]

   if -4.2000000000000002e-4 < F < -4.8e-151

   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

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

      1. /-lowering-/.f64 79.9%

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

   5. Simplified 79.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

   7. Applied egg-rr 79.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 79.9%

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

   if -4.8e-151 < F < 0.14000000000000001

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

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

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

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

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 85.4%

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

   9. Simplified 85.4%

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

   if 0.14000000000000001 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00042:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-151}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-151}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.124:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00049)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.8e-151)
       (- (* F (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B))) (/ x B))
       (if (<= F 0.124)
         (- (/ F (* B (sqrt (+ (* x 2.0) (+ (* F F) 2.0))))) t_0)
         (- (/ 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 <= -0.00049) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.8e-151) {
		tmp = (F * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - (x / B);
	} else if (F <= 0.124) {
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= (-0.00049d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.8d-151)) then
        tmp = (f * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - (x / b)
    else if (f <= 0.124d0) then
        tmp = (f / (b * sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))))) - t_0
    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 <= -0.00049) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.8e-151) {
		tmp = (F * (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - (x / B);
	} else if (F <= 0.124) {
		tmp = (F / (B * Math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= -0.00049:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.8e-151:
		tmp = (F * (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - (x / B)
	elif F <= 0.124:
		tmp = (F / (B * math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0
	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 <= -0.00049)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.8e-151)
		tmp = Float64(Float64(F * Float64((Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5) / sin(B))) - Float64(x / B));
	elseif (F <= 0.124)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))))) - t_0);
	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 <= -0.00049)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.8e-151)
		tmp = (F * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - (x / B);
	elseif (F <= 0.124)
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	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, -0.00049], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.8e-151], N[(N[(F * N[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.124], N[(N[(F / N[(B * N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.8999999999999998e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -4.8999999999999998e-4 < F < -4.8e-151

   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

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

      1. /-lowering-/.f64 79.9%

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

   5. Simplified 79.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

   7. Applied egg-rr 79.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 79.9%

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

   if -4.8e-151 < F < 0.124

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

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

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

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

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 85.4%

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

   9. Simplified 85.4%

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

   if 0.124 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-151}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.124:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00049)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4e-147)
       (- (/ (/ F (sin B)) (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) (/ x B))
       (if (<= F 0.14)
         (- (/ F (* B (sqrt (+ (* x 2.0) (+ (* F F) 2.0))))) t_0)
         (- (/ 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 <= -0.00049) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4e-147) {
		tmp = ((F / sin(B)) / sqrt(((F * F) + (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= (-0.00049d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4d-147)) then
        tmp = ((f / sin(b)) / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = (f / (b * sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))))) - t_0
    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 <= -0.00049) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4e-147) {
		tmp = ((F / Math.sin(B)) / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * Math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= -0.00049:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4e-147:
		tmp = ((F / math.sin(B)) / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 0.14:
		tmp = (F / (B * math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0
	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 <= -0.00049)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4e-147)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 0.14)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))))) - t_0);
	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 <= -0.00049)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4e-147)
		tmp = ((F / sin(B)) / sqrt(((F * F) + (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 0.14)
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	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, -0.00049], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4e-147], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F / N[(B * N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.8999999999999998e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -4.8999999999999998e-4 < F < -3.9999999999999999e-147

   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

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

      1. /-lowering-/.f64 79.3%

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

   5. Simplified 79.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

   7. Applied egg-rr 79.2%

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

      1. associate-/r* N/A

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

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

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

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

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

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

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 79.3%

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

   9. Applied egg-rr 79.3%

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

   if -3.9999999999999999e-147 < F < 0.14000000000000001

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

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

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

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

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 85.5%

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

   9. Simplified 85.5%

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

   if 0.14000000000000001 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.5e-6)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4e-147)
       (- (/ F (* (sin B) (sqrt (+ (* F F) (+ 2.0 (* x 2.0)))))) (/ x B))
       (if (<= F 0.14)
         (- (/ F (* B (sqrt (+ (* x 2.0) (+ (* F F) 2.0))))) t_0)
         (- (/ 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 <= -9.5e-6) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4e-147) {
		tmp = (F / (sin(B) * sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= (-9.5d-6)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4d-147)) then
        tmp = (f / (sin(b) * sqrt(((f * f) + (2.0d0 + (x * 2.0d0)))))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = (f / (b * sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))))) - t_0
    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 <= -9.5e-6) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4e-147) {
		tmp = (F / (Math.sin(B) * Math.sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * Math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= -9.5e-6:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4e-147:
		tmp = (F / (math.sin(B) * math.sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B)
	elif F <= 0.14:
		tmp = (F / (B * math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0
	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 <= -9.5e-6)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4e-147)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0)))))) - Float64(x / B));
	elseif (F <= 0.14)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))))) - t_0);
	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 <= -9.5e-6)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4e-147)
		tmp = (F / (sin(B) * sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B);
	elseif (F <= 0.14)
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	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, -9.5e-6], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4e-147], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F / N[(B * N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.5000000000000005e-6

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -9.5000000000000005e-6 < F < -3.9999999999999999e-147

   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

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

      1. /-lowering-/.f64 79.3%

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

   5. Simplified 79.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

   7. Applied egg-rr 79.2%

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

   if -3.9999999999999999e-147 < F < 0.14000000000000001

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

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

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

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

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 85.5%

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

   9. Simplified 85.5%

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

   if 0.14000000000000001 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00044:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00044)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.8e-151)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (if (<= F 0.14)
         (- (/ F (* B (sqrt (+ (* x 2.0) (+ (* F F) 2.0))))) t_0)
         (- (/ 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 <= -0.00044) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.8e-151) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= (-0.00044d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.8d-151)) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = (f / (b * sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))))) - t_0
    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 <= -0.00044) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.8e-151) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (F / (B * Math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	} 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 <= -0.00044:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.8e-151:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 0.14:
		tmp = (F / (B * math.sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0
	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 <= -0.00044)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.8e-151)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 0.14)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))))) - t_0);
	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 <= -0.00044)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.8e-151)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 0.14)
		tmp = (F / (B * sqrt(((x * 2.0) + ((F * F) + 2.0))))) - t_0;
	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, -0.00044], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.8e-151], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F / N[(B * N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.40000000000000016e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -4.40000000000000016e-4 < F < -4.8e-151

   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

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

      1. /-lowering-/.f64 79.9%

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

   5. Simplified 79.9%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 79.4%

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

   8. Simplified 79.4%

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

   if -4.8e-151 < F < 0.14000000000000001

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

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

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

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

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 85.4%

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

   9. Simplified 85.4%

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

   if 0.14000000000000001 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00044:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{F \cdot F + 2}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00028:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (+ (* F F) 2.0)))))
        (t_1 (/ x (tan B))))
   (if (<= F -0.00028)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4e-114)
       t_0
       (if (<= F 2.6e-135)
         (/ (* x (cos B)) (- 0.0 (sin B)))
         (if (<= F 5.8e-5) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / sin(B)) * sqrt((1.0 / ((F * F) + 2.0)));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.00028) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4e-114) {
		tmp = t_0;
	} else if (F <= 2.6e-135) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 5.8e-5) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	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)) * sqrt((1.0d0 / ((f * f) + 2.0d0)))
    t_1 = x / tan(b)
    if (f <= (-0.00028d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4d-114)) then
        tmp = t_0
    else if (f <= 2.6d-135) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 5.8d-5) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    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.sqrt((1.0 / ((F * F) + 2.0)));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00028) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4e-114) {
		tmp = t_0;
	} else if (F <= 2.6e-135) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 5.8e-5) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / math.sin(B)) * math.sqrt((1.0 / ((F * F) + 2.0)))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.00028:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4e-114:
		tmp = t_0
	elif F <= 2.6e-135:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 5.8e-5:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0))))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00028)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4e-114)
		tmp = t_0;
	elseif (F <= 2.6e-135)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 5.8e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / sin(B)) * sqrt((1.0 / ((F * F) + 2.0)));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00028)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4e-114)
		tmp = t_0;
	elseif (F <= 2.6e-135)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 5.8e-5)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00028], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4e-114], t$95$0, If[LessEqual[F, 2.6e-135], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-5], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7999999999999998e-4

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 79.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

   if -2.7999999999999998e-4 < F < -4.0000000000000002e-114 or 2.60000000000000004e-135 < F < 5.8e-5

   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 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 66.0%

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

   5. Simplified 66.0%

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

   if -4.0000000000000002e-114 < F < 2.60000000000000004e-135

   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(0 - \color{blue}{\sin B}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\sin B}\right)\right) \]

      8. sin-lowering-sin.f64 83.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{0 - \sin B}} \]

   if 5.8e-5 < F

   1. Initial program 52.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00028:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-114}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{F \cdot F + 2}}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00066:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.3e-29)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.8e-93)
       (/ (* x (cos B)) (- 0.0 (sin B)))
       (if (<= F 0.00066)
         (-
          (/
           F
           (*
            B
            (*
             (sqrt (+ (* x 2.0) (+ (* F F) 2.0)))
             (+ 1.0 (* (* B B) -0.16666666666666666)))))
          (/ 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 <= -3.3e-29) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.8e-93) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 0.00066) {
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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 <= (-3.3d-29)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.8d-93) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 0.00066d0) then
        tmp = (f / (b * (sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))) * (1.0d0 + ((b * b) * (-0.16666666666666666d0)))))) - (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 <= -3.3e-29) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.8e-93) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 0.00066) {
		tmp = (F / (B * (Math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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 <= -3.3e-29:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.8e-93:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 0.00066:
		tmp = (F / (B * (math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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 <= -3.3e-29)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.8e-93)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 0.00066)
		tmp = Float64(Float64(F / Float64(B * Float64(sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))) * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666))))) - Float64(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 <= -3.3e-29)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.8e-93)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 0.00066)
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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, -3.3e-29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e-93], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00066], N[(N[(F / N[(B * N[(N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.30000000000000028e-29

   1. Initial program 70.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

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

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

   4. Applied egg-rr 80.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 92.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -3.30000000000000028e-29 < F < 5.7999999999999997e-93

   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. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{x \cdot \cos B}{\color{blue}{\mathsf{neg}\left(\sin B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(0 - \color{blue}{\sin B}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\sin B}\right)\right) \]

      8. sin-lowering-sin.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{0 - \sin B}} \]

   if 5.7999999999999997e-93 < F < 6.6e-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. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \left(\frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(\frac{-1}{6} \cdot {B}^{2} + 1\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 + 2 \cdot x\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot x + 2\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot x + \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 57.1%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \left(\left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right) \cdot \sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}} - \frac{x}{B} \]

   if 6.6e-4 < F

   1. Initial program 52.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\color{blue}{\tan B}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.00066:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-93}:\\ \;\;\;\;\frac{\cos B}{\sin B} \cdot \left(0 - x\right)\\ \mathbf{elif}\;F \leq 0.011:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.3e-30)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7e-93)
       (* (/ (cos B) (sin B)) (- 0.0 x))
       (if (<= F 0.011)
         (-
          (/
           F
           (*
            B
            (*
             (sqrt (+ (* x 2.0) (+ (* F F) 2.0)))
             (+ 1.0 (* (* B B) -0.16666666666666666)))))
          (/ 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 <= -5.3e-30) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7e-93) {
		tmp = (cos(B) / sin(B)) * (0.0 - x);
	} else if (F <= 0.011) {
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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 <= (-5.3d-30)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 7d-93) then
        tmp = (cos(b) / sin(b)) * (0.0d0 - x)
    else if (f <= 0.011d0) then
        tmp = (f / (b * (sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))) * (1.0d0 + ((b * b) * (-0.16666666666666666d0)))))) - (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 <= -5.3e-30) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 7e-93) {
		tmp = (Math.cos(B) / Math.sin(B)) * (0.0 - x);
	} else if (F <= 0.011) {
		tmp = (F / (B * (Math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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 <= -5.3e-30:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 7e-93:
		tmp = (math.cos(B) / math.sin(B)) * (0.0 - x)
	elif F <= 0.011:
		tmp = (F / (B * (math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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 <= -5.3e-30)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7e-93)
		tmp = Float64(Float64(cos(B) / sin(B)) * Float64(0.0 - x));
	elseif (F <= 0.011)
		tmp = Float64(Float64(F / Float64(B * Float64(sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))) * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666))))) - Float64(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 <= -5.3e-30)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 7e-93)
		tmp = (cos(B) / sin(B)) * (0.0 - x);
	elseif (F <= 0.011)
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (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, -5.3e-30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7e-93], N[(N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(0.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.011], N[(N[(F / N[(B * N[(N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.29999999999999974e-30

   1. Initial program 70.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(\frac{1}{\sin B} \cdot \color{blue}{F}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot \color{blue}{F}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \cdot F\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \color{blue}{F}\right)\right) \]

   4. Applied egg-rr 80.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sin B}{F}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 92.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -5.29999999999999974e-30 < F < 7e-93

   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. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(\frac{1}{\sin B} \cdot \color{blue}{F}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot \color{blue}{F}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \cdot F\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \color{blue}{F}\right)\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sin B}{F}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   8. 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 72.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\left(0 - x\right) \cdot \frac{\cos B}{\sin B}} \]

   if 7e-93 < F < 0.010999999999999999

   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. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \left(\frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(\frac{-1}{6} \cdot {B}^{2} + 1\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 + 2 \cdot x\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot x + 2\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot x + \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 57.1%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \left(\left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right) \cdot \sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}} - \frac{x}{B} \]

   if 0.010999999999999999 < F

   1. Initial program 52.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\color{blue}{x} \cdot \frac{1}{\tan B}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{\color{blue}{\tan B}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\tan B}\right)\right) \]

      8. tan-lowering-tan.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. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-93}:\\ \;\;\;\;\frac{\cos B}{\sin B} \cdot \left(0 - x\right)\\ \mathbf{elif}\;F \leq 0.011:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{\cos B}{\sin B} \cdot \left(0 - x\right)\\ \mathbf{elif}\;F \leq 0.066:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-28)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2.7e-93)
     (* (/ (cos B) (sin B)) (- 0.0 x))
     (if (<= F 0.066)
       (-
        (/
         F
         (*
          B
          (*
           (sqrt (+ (* x 2.0) (+ (* F F) 2.0)))
           (+ 1.0 (* (* B B) -0.16666666666666666)))))
        (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-28) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.7e-93) {
		tmp = (cos(B) / sin(B)) * (0.0 - x);
	} else if (F <= 0.066) {
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 <= (-1.3d-28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 2.7d-93) then
        tmp = (cos(b) / sin(b)) * (0.0d0 - x)
    else if (f <= 0.066d0) then
        tmp = (f / (b * (sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))) * (1.0d0 + ((b * b) * (-0.16666666666666666d0)))))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-28) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 2.7e-93) {
		tmp = (Math.cos(B) / Math.sin(B)) * (0.0 - x);
	} else if (F <= 0.066) {
		tmp = (F / (B * (Math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e-28:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 2.7e-93:
		tmp = (math.cos(B) / math.sin(B)) * (0.0 - x)
	elif F <= 0.066:
		tmp = (F / (B * (math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.7e-93)
		tmp = Float64(Float64(cos(B) / sin(B)) * Float64(0.0 - x));
	elseif (F <= 0.066)
		tmp = Float64(Float64(F / Float64(B * Float64(sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))) * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666))))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e-28)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 2.7e-93)
		tmp = (cos(B) / sin(B)) * (0.0 - x);
	elseif (F <= 0.066)
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-93], N[(N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(0.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.066], N[(N[(F / N[(B * N[(N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3e-28

   1. Initial program 70.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(\frac{1}{\sin B} \cdot \color{blue}{F}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot \color{blue}{F}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \cdot F\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \color{blue}{F}\right)\right) \]

   4. Applied egg-rr 80.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sin B}{F}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   6. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. 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 92.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -1.3e-28 < F < 2.7000000000000001e-93

   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. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(\frac{1}{\sin B} \cdot \color{blue}{F}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot \color{blue}{F}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \cdot F\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \color{blue}{F}\right)\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sin B}{F}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   8. 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 72.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\left(0 - x\right) \cdot \frac{\cos B}{\sin B}} \]

   if 2.7000000000000001e-93 < F < 0.066000000000000003

   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. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 83.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \left(\frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(\frac{-1}{6} \cdot {B}^{2} + 1\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 + 2 \cdot x\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot x + 2\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot x + \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 57.1%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \left(\left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right) \cdot \sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}} - \frac{x}{B} \]

   if 0.066000000000000003 < F

   1. Initial program 52.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 78.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 78.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{\cos B}{\sin B} \cdot \left(0 - x\right)\\ \mathbf{elif}\;F \leq 0.066:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.0095:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F \cdot \left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right)\right) + \left(B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos B}{\sin B} \cdot \left(0 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.0095)
   (/
    (+
     (*
      (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
      (* F (+ 1.0 (* (* B B) 0.16666666666666666))))
     (- (* B (* B (* x 0.3333333333333333))) x))
    B)
   (* (/ (cos B) (sin B)) (- 0.0 x))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.0095) {
		tmp = ((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B;
	} else {
		tmp = (cos(B) / sin(B)) * (0.0 - x);
	}
	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 <= 0.0095d0) then
        tmp = ((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f * (1.0d0 + ((b * b) * 0.16666666666666666d0)))) + ((b * (b * (x * 0.3333333333333333d0))) - x)) / b
    else
        tmp = (cos(b) / sin(b)) * (0.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.0095) {
		tmp = ((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B;
	} else {
		tmp = (Math.cos(B) / Math.sin(B)) * (0.0 - x);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.0095:
		tmp = ((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B
	else:
		tmp = (math.cos(B) / math.sin(B)) * (0.0 - x)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.0095)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F * Float64(1.0 + Float64(Float64(B * B) * 0.16666666666666666)))) + Float64(Float64(B * Float64(B * Float64(x * 0.3333333333333333))) - x)) / B);
	else
		tmp = Float64(Float64(cos(B) / sin(B)) * Float64(0.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.0095)
		tmp = ((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B;
	else
		tmp = (cos(B) / sin(B)) * (0.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.0095], 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[(1.0 + N[(N[(B * B), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00949999999999999976

   1. Initial program 75.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(\left(0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right) \cdot F\right) + \left(B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right) - x\right)}{B}} \]

   if 0.00949999999999999976 < B

   1. Initial program 87.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(\frac{1}{\sin B} \cdot \color{blue}{F}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot \color{blue}{F}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \cdot F\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \color{blue}{F}\right)\right) \]

   4. Applied egg-rr 87.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} \cdot F} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sin B}{F}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   6. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   8. 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 64.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(B\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 64.0%

      \[\leadsto \color{blue}{\left(0 - x\right) \cdot \frac{\cos B}{\sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0095:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F \cdot \left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right)\right) + \left(B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos B}{\sin B} \cdot \left(0 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1050000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F \cdot \left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right)\right) + \left(B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1050000.0)
   (/
    (+
     (*
      (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
      (* F (+ 1.0 (* (* B B) 0.16666666666666666))))
     (- (* B (* B (* x 0.3333333333333333))) x))
    B)
   (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1050000.0) {
		tmp = ((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (b <= 1050000.0d0) then
        tmp = ((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f * (1.0d0 + ((b * b) * 0.16666666666666666d0)))) + ((b * (b * (x * 0.3333333333333333d0))) - x)) / b
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1050000.0) {
		tmp = ((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1050000.0:
		tmp = ((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1050000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F * Float64(1.0 + Float64(Float64(B * B) * 0.16666666666666666)))) + Float64(Float64(B * Float64(B * Float64(x * 0.3333333333333333))) - x)) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1050000.0)
		tmp = ((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + ((B * (B * (x * 0.3333333333333333))) - x)) / B;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1050000.0], 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[(1.0 + N[(N[(B * B), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.05e6

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(\left(0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right) \cdot F\right) + \left(B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right) - x\right)}{B}} \]

   if 1.05e6 < B

   1. Initial program 87.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 56.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 62.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1050000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F \cdot \left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right)\right) + \left(B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{F \cdot \left(0 - \sin B\right)} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-165}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}}}{B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ F (* F (- 0.0 (sin B)))) (/ x B)))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))
   (if (<= x -8.8e-10)
     t_1
     (if (<= x -3.3e-165)
       t_0
       (if (<= x 3.3e-245)
         (/ (* F (sqrt (/ 1.0 (+ (* F F) 2.0)))) B)
         (if (<= x 5.8e-59) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / (F * (0.0 - sin(B)))) - (x / B);
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -8.8e-10) {
		tmp = t_1;
	} else if (x <= -3.3e-165) {
		tmp = t_0;
	} else if (x <= 3.3e-245) {
		tmp = (F * sqrt((1.0 / ((F * F) + 2.0)))) / B;
	} else if (x <= 5.8e-59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 / (f * (0.0d0 - sin(b)))) - (x / b)
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    if (x <= (-8.8d-10)) then
        tmp = t_1
    else if (x <= (-3.3d-165)) then
        tmp = t_0
    else if (x <= 3.3d-245) then
        tmp = (f * sqrt((1.0d0 / ((f * f) + 2.0d0)))) / b
    else if (x <= 5.8d-59) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F / (F * (0.0 - Math.sin(B)))) - (x / B);
	double t_1 = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -8.8e-10) {
		tmp = t_1;
	} else if (x <= -3.3e-165) {
		tmp = t_0;
	} else if (x <= 3.3e-245) {
		tmp = (F * Math.sqrt((1.0 / ((F * F) + 2.0)))) / B;
	} else if (x <= 5.8e-59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / (F * (0.0 - math.sin(B)))) - (x / B)
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	tmp = 0
	if x <= -8.8e-10:
		tmp = t_1
	elif x <= -3.3e-165:
		tmp = t_0
	elif x <= 3.3e-245:
		tmp = (F * math.sqrt((1.0 / ((F * F) + 2.0)))) / B
	elif x <= 5.8e-59:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / Float64(F * Float64(0.0 - sin(B)))) - Float64(x / B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -8.8e-10)
		tmp = t_1;
	elseif (x <= -3.3e-165)
		tmp = t_0;
	elseif (x <= 3.3e-245)
		tmp = Float64(Float64(F * sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0)))) / B);
	elseif (x <= 5.8e-59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / (F * (0.0 - sin(B)))) - (x / B);
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	tmp = 0.0;
	if (x <= -8.8e-10)
		tmp = t_1;
	elseif (x <= -3.3e-165)
		tmp = t_0;
	elseif (x <= 3.3e-245)
		tmp = (F * sqrt((1.0 / ((F * F) + 2.0)))) / B;
	elseif (x <= 5.8e-59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[(F * N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.8e-10], t$95$1, If[LessEqual[x, -3.3e-165], t$95$0, If[LessEqual[x, 3.3e-245], N[(N[(F * N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 5.8e-59], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.7999999999999996e-10 or 5.80000000000000033e-59 < 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. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 87.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 87.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 92.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

   if -8.7999999999999996e-10 < x < -3.2999999999999998e-165 or 3.3000000000000001e-245 < x < 5.80000000000000033e-59

   1. Initial program 71.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 52.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 52.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(-1 \cdot \left(F \cdot \sin B\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\mathsf{neg}\left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(0 - F \cdot \sin B\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      6. sin-lowering-sin.f64 35.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 35.2%

       \[\leadsto \frac{F}{\color{blue}{0 - \sin B \cdot F}} - \frac{x}{B} \]

   if -3.2999999999999998e-165 < x < 3.3000000000000001e-245

   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 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 x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, B\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), B\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), B\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), B\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), B\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), B\right) \]

      6. *-lowering-*.f64 41.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), B\right) \]

   8. Simplified 41.8%

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{F}{F \cdot \left(0 - \sin B\right)} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}}}{B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{F}{F \cdot \left(0 - \sin B\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{F \cdot \left(0 - \sin B\right)} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;x \leq -1.66 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ F (* F (- 0.0 (sin B)))) (/ x B)))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))
   (if (<= x -1.66e-10)
     t_1
     (if (<= x -2.8e-166)
       t_0
       (if (<= x 8.5e-246)
         (* (sqrt (/ 1.0 (+ (* F F) 2.0))) (/ F B))
         (if (<= x 4.8e-60) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / (F * (0.0 - sin(B)))) - (x / B);
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -1.66e-10) {
		tmp = t_1;
	} else if (x <= -2.8e-166) {
		tmp = t_0;
	} else if (x <= 8.5e-246) {
		tmp = sqrt((1.0 / ((F * F) + 2.0))) * (F / B);
	} else if (x <= 4.8e-60) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 / (f * (0.0d0 - sin(b)))) - (x / b)
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    if (x <= (-1.66d-10)) then
        tmp = t_1
    else if (x <= (-2.8d-166)) then
        tmp = t_0
    else if (x <= 8.5d-246) then
        tmp = sqrt((1.0d0 / ((f * f) + 2.0d0))) * (f / b)
    else if (x <= 4.8d-60) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F / (F * (0.0 - Math.sin(B)))) - (x / B);
	double t_1 = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -1.66e-10) {
		tmp = t_1;
	} else if (x <= -2.8e-166) {
		tmp = t_0;
	} else if (x <= 8.5e-246) {
		tmp = Math.sqrt((1.0 / ((F * F) + 2.0))) * (F / B);
	} else if (x <= 4.8e-60) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / (F * (0.0 - math.sin(B)))) - (x / B)
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	tmp = 0
	if x <= -1.66e-10:
		tmp = t_1
	elif x <= -2.8e-166:
		tmp = t_0
	elif x <= 8.5e-246:
		tmp = math.sqrt((1.0 / ((F * F) + 2.0))) * (F / B)
	elif x <= 4.8e-60:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / Float64(F * Float64(0.0 - sin(B)))) - Float64(x / B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -1.66e-10)
		tmp = t_1;
	elseif (x <= -2.8e-166)
		tmp = t_0;
	elseif (x <= 8.5e-246)
		tmp = Float64(sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0))) * Float64(F / B));
	elseif (x <= 4.8e-60)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / (F * (0.0 - sin(B)))) - (x / B);
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	tmp = 0.0;
	if (x <= -1.66e-10)
		tmp = t_1;
	elseif (x <= -2.8e-166)
		tmp = t_0;
	elseif (x <= 8.5e-246)
		tmp = sqrt((1.0 / ((F * F) + 2.0))) * (F / B);
	elseif (x <= 4.8e-60)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[(F * N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.66e-10], t$95$1, If[LessEqual[x, -2.8e-166], t$95$0, If[LessEqual[x, 8.5e-246], N[(N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-60], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.66e-10 or 4.80000000000000019e-60 < 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. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 87.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 87.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 92.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

   if -1.66e-10 < x < -2.7999999999999999e-166 or 8.4999999999999998e-246 < x < 4.80000000000000019e-60

   1. Initial program 71.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 52.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 52.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(-1 \cdot \left(F \cdot \sin B\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\mathsf{neg}\left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(0 - F \cdot \sin B\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      6. sin-lowering-sin.f64 35.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 35.2%

       \[\leadsto \frac{F}{\color{blue}{0 - \sin B \cdot F}} - \frac{x}{B} \]

   if -2.7999999999999999e-166 < x < 8.4999999999999998e-246

   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 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 x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{B}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\left(\frac{F}{B}\right)}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{\color{blue}{F}}{B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      8. /-lowering-/.f64 39.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, \color{blue}{B}\right)\right) \]

   8. Simplified 39.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{F}{F \cdot \left(0 - \sin B\right)} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{F}{F \cdot \left(0 - \sin B\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 7800000:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 7800000.0)
   (-
    (/
     F
     (*
      B
      (*
       (sqrt (+ (* x 2.0) (+ (* F F) 2.0)))
       (+ 1.0 (* (* B B) -0.16666666666666666)))))
    (/ x B))
   (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 7800000.0) {
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (b <= 7800000.0d0) then
        tmp = (f / (b * (sqrt(((x * 2.0d0) + ((f * f) + 2.0d0))) * (1.0d0 + ((b * b) * (-0.16666666666666666d0)))))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 7800000.0) {
		tmp = (F / (B * (Math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 7800000.0:
		tmp = (F / (B * (math.sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 7800000.0)
		tmp = Float64(Float64(F / Float64(B * Float64(sqrt(Float64(Float64(x * 2.0) + Float64(Float64(F * F) + 2.0))) * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666))))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 7800000.0)
		tmp = (F / (B * (sqrt(((x * 2.0) + ((F * F) + 2.0))) * (1.0 + ((B * B) * -0.16666666666666666))))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 7800000.0], N[(N[(F / N[(B * N[(N[Sqrt[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.8e6

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 51.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 51.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \frac{-1}{6} \cdot \left({B}^{2} \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} + \left(\frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(\frac{-1}{6} \cdot {B}^{2} + 1\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {B}^{2}\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \left(\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 + 2 \cdot x\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot x + 2\right) + {F}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot x + \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(2 + {F}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 52.9%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \left(\left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right) \cdot \sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}} - \frac{x}{B} \]

   if 7.8e6 < B

   1. Initial program 87.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 56.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 62.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7800000:\\ \;\;\;\;\frac{F}{B \cdot \left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)} \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-165}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B)))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))
   (if (<= x -2.9e-10)
     t_1
     (if (<= x -3.6e-165)
       t_0
       (if (<= x 6.6e-246)
         (* (sqrt (/ 1.0 (+ (* F F) 2.0))) (/ F B))
         (if (<= x 1.9e-58) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -2.9e-10) {
		tmp = t_1;
	} else if (x <= -3.6e-165) {
		tmp = t_0;
	} else if (x <= 6.6e-246) {
		tmp = sqrt((1.0 / ((F * F) + 2.0))) * (F / B);
	} else if (x <= 1.9e-58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = ((-1.0d0) / sin(b)) - (x / b)
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    if (x <= (-2.9d-10)) then
        tmp = t_1
    else if (x <= (-3.6d-165)) then
        tmp = t_0
    else if (x <= 6.6d-246) then
        tmp = sqrt((1.0d0 / ((f * f) + 2.0d0))) * (f / b)
    else if (x <= 1.9d-58) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / Math.sin(B)) - (x / B);
	double t_1 = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -2.9e-10) {
		tmp = t_1;
	} else if (x <= -3.6e-165) {
		tmp = t_0;
	} else if (x <= 6.6e-246) {
		tmp = Math.sqrt((1.0 / ((F * F) + 2.0))) * (F / B);
	} else if (x <= 1.9e-58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	tmp = 0
	if x <= -2.9e-10:
		tmp = t_1
	elif x <= -3.6e-165:
		tmp = t_0
	elif x <= 6.6e-246:
		tmp = math.sqrt((1.0 / ((F * F) + 2.0))) * (F / B)
	elif x <= 1.9e-58:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -2.9e-10)
		tmp = t_1;
	elseif (x <= -3.6e-165)
		tmp = t_0;
	elseif (x <= 6.6e-246)
		tmp = Float64(sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0))) * Float64(F / B));
	elseif (x <= 1.9e-58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / sin(B)) - (x / B);
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	tmp = 0.0;
	if (x <= -2.9e-10)
		tmp = t_1;
	elseif (x <= -3.6e-165)
		tmp = t_0;
	elseif (x <= 6.6e-246)
		tmp = sqrt((1.0 / ((F * F) + 2.0))) * (F / B);
	elseif (x <= 1.9e-58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e-10], t$95$1, If[LessEqual[x, -3.6e-165], t$95$0, If[LessEqual[x, 6.6e-246], N[(N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-58], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.89999999999999981e-10 or 1.8999999999999999e-58 < 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. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 87.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 87.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 92.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

   if -2.89999999999999981e-10 < x < -3.59999999999999984e-165 or 6.6000000000000002e-246 < x < 1.8999999999999999e-58

   1. Initial program 71.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 52.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 52.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 35.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 35.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -3.59999999999999984e-165 < x < 6.6000000000000002e-246

   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 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 x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{B}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\left(\frac{F}{B}\right)}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{\color{blue}{F}}{B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      8. /-lowering-/.f64 39.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, \color{blue}{B}\right)\right) \]

   8. Simplified 39.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 55.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1050000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1050000.0)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1050000.0) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (b <= 1050000.0d0) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1050000.0) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1050000.0:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1050000.0)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1050000.0)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1050000.0], N[(N[(N[(F / N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.05e6

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 52.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   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 52.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}} \]

   if 1.05e6 < B

   1. Initial program 87.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 56.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 62.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1050000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-66)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.3e-45) (- 0.0 (/ x B)) (- (/ F (* F (sin B))) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-66) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.3e-45) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (F / (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 <= (-2.4d-66)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.3d-45) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = (f / (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 <= -2.4e-66) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.3e-45) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-66:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.3e-45:
		tmp = 0.0 - (x / B)
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-66)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.3e-45)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e-66)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.3e-45)
		tmp = 0.0 - (x / B);
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-66], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-45], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.40000000000000026e-66

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 39.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 39.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 56.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -2.40000000000000026e-66 < F < 1.29999999999999993e-45

   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 40.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 40.7%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 27.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 27.2%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 1.29999999999999993e-45 < F

   1. Initial program 58.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 30.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 30.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      7. clear-num N/A

         \[\leadsto {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\frac{x}{\color{blue}{B}}\right)\right) \]

      8. div-inv N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \color{blue}{\frac{x}{B}} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

   7. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(F \cdot \sin B\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\sin B \cdot F\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\sin B, F\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

      3. sin-lowering-sin.f64 62.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   10. Simplified 62.3%

       \[\leadsto \frac{F}{\color{blue}{\sin B \cdot F}} - \frac{x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 55.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1050000:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1050000.0)
   (/ (- (* F (sqrt (/ 1.0 (+ (* F F) 2.0)))) x) B)
   (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1050000.0) {
		tmp = ((F * sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (b <= 1050000.0d0) then
        tmp = ((f * sqrt((1.0d0 / ((f * f) + 2.0d0)))) - x) / b
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1050000.0) {
		tmp = ((F * Math.sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1050000.0:
		tmp = ((F * math.sqrt((1.0 / ((F * F) + 2.0)))) - x) / B
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1050000.0)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1050000.0)
		tmp = ((F * sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1050000.0], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.05e6

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 52.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), x\right), B\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), x\right), B\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), x\right), B\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), x\right), B\right) \]

      6. *-lowering-*.f64 52.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), x\right), B\right) \]

   8. Simplified 52.5%

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}} - x}{B} \]

   if 1.05e6 < B

   1. Initial program 87.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 56.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 62.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1050000:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 62.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.25e+17)
   (- (/ -1.0 (sin B)) (/ x B))
   (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.25e+17) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 <= (-2.25d+17)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.25e+17) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.25e+17:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.25e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.25e+17)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.25e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.25e17

   1. Initial program 64.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 33.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 33.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 67.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -2.25e17 < F

   1. Initial program 85.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 55.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 55.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{B}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 57.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right) \]

   8. Simplified 57.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 44.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+218}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot -0.00205026455026455 + -0.019444444444444445\right)\right) + \left(-1 - x\right)}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-26}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e+218)
   (/
    (+
     (*
      (* B B)
      (+
       -0.16666666666666666
       (* (* B B) (+ (* (* B B) -0.00205026455026455) -0.019444444444444445))))
     (- -1.0 x))
    B)
   (if (<= F -3.9e-50)
     (/ -1.0 (sin B))
     (if (<= F 9.2e-26)
       (- 0.0 (/ x B))
       (/
        (+
         (* (* B B) (+ 0.16666666666666666 (* x 0.3333333333333333)))
         (- 1.0 x))
        B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e+218) {
		tmp = (((B * B) * (-0.16666666666666666 + ((B * B) * (((B * B) * -0.00205026455026455) + -0.019444444444444445)))) + (-1.0 - x)) / B;
	} else if (F <= -3.9e-50) {
		tmp = -1.0 / sin(B);
	} else if (F <= 9.2e-26) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (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 <= (-2.5d+218)) then
        tmp = (((b * b) * ((-0.16666666666666666d0) + ((b * b) * (((b * b) * (-0.00205026455026455d0)) + (-0.019444444444444445d0))))) + ((-1.0d0) - x)) / b
    else if (f <= (-3.9d-50)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 9.2d-26) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = (((b * b) * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (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 <= -2.5e+218) {
		tmp = (((B * B) * (-0.16666666666666666 + ((B * B) * (((B * B) * -0.00205026455026455) + -0.019444444444444445)))) + (-1.0 - x)) / B;
	} else if (F <= -3.9e-50) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 9.2e-26) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e+218:
		tmp = (((B * B) * (-0.16666666666666666 + ((B * B) * (((B * B) * -0.00205026455026455) + -0.019444444444444445)))) + (-1.0 - x)) / B
	elif F <= -3.9e-50:
		tmp = -1.0 / math.sin(B)
	elif F <= 9.2e-26:
		tmp = 0.0 - (x / B)
	else:
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e+218)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(-0.16666666666666666 + Float64(Float64(B * B) * Float64(Float64(Float64(B * B) * -0.00205026455026455) + -0.019444444444444445)))) + Float64(-1.0 - x)) / B);
	elseif (F <= -3.9e-50)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 9.2e-26)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 - x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e+218)
		tmp = (((B * B) * (-0.16666666666666666 + ((B * B) * (((B * B) * -0.00205026455026455) + -0.019444444444444445)))) + (-1.0 - x)) / B;
	elseif (F <= -3.9e-50)
		tmp = -1.0 / sin(B);
	elseif (F <= 9.2e-26)
		tmp = 0.0 - (x / B);
	else
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e+218], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * -0.00205026455026455), $MachinePrecision] + -0.019444444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.9e-50], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.2e-26], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.49999999999999991e218

   1. Initial program 37.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 8.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 8.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 71.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - \left(1 + x\right)}{B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - \left(1 + x\right)\right), \color{blue}{B}\right) \]

   11. Simplified 56.8%

       \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot -0.00205026455026455 + -0.019444444444444445\right) + -0.16666666666666666\right) + \left(-1 - x\right)}{B}} \]

   if -2.49999999999999991e218 < F < -3.90000000000000021e-50

   1. Initial program 86.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 50.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 50.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 49.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   9. Taylor expanded in B around inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\sin B}\right) \]

       2. sin-lowering-sin.f64 40.6%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right) \]

   11. Simplified 40.6%

       \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -3.90000000000000021e-50 < F < 9.20000000000000035e-26

   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 39.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.8%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 9.20000000000000035e-26 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) - x\right), B\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 - x\right)\right), B\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{3} \cdot x\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      11. --lowering--.f64 48.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), B\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+218}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot -0.00205026455026455 + -0.019444444444444445\right)\right) + \left(-1 - x\right)}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-26}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 51.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.05e-66)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 6.4e-22)
     (- 0.0 (/ x B))
     (/
      (+
       (* (* B B) (+ 0.16666666666666666 (* x 0.3333333333333333)))
       (- 1.0 x))
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.05e-66) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 6.4e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (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 <= (-2.05d-66)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 6.4d-22) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = (((b * b) * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (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 <= -2.05e-66) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 6.4e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.05e-66:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 6.4e-22:
		tmp = 0.0 - (x / B)
	else:
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.05e-66)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 6.4e-22)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 - x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.05e-66)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 6.4e-22)
		tmp = 0.0 - (x / B);
	else
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.05e-66], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.4e-22], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.04999999999999999e-66

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 39.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 39.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 56.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -2.04999999999999999e-66 < F < 6.39999999999999975e-22

   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 39.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.3%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 6.39999999999999975e-22 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) - x\right), B\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 - x\right)\right), B\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{3} \cdot x\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      11. --lowering--.f64 48.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), B\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 44.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(-0.16666666666666666 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right)\right)\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.6e-48)
   (-
    (/
     -1.0
     (*
      B
      (+
       1.0
       (*
        (* B B)
        (+
         -0.16666666666666666
         (*
          (* B B)
          (+ 0.008333333333333333 (* (* B B) -0.0001984126984126984))))))))
    (/ x B))
   (if (<= F 2.05e-26)
     (- 0.0 (/ x B))
     (/
      (+
       (* (* B B) (+ 0.16666666666666666 (* x 0.3333333333333333)))
       (- 1.0 x))
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.6e-48) {
		tmp = (-1.0 / (B * (1.0 + ((B * B) * (-0.16666666666666666 + ((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984)))))))) - (x / B);
	} else if (F <= 2.05e-26) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (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 <= (-5.6d-48)) then
        tmp = ((-1.0d0) / (b * (1.0d0 + ((b * b) * ((-0.16666666666666666d0) + ((b * b) * (0.008333333333333333d0 + ((b * b) * (-0.0001984126984126984d0))))))))) - (x / b)
    else if (f <= 2.05d-26) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = (((b * b) * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (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 <= -5.6e-48) {
		tmp = (-1.0 / (B * (1.0 + ((B * B) * (-0.16666666666666666 + ((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984)))))))) - (x / B);
	} else if (F <= 2.05e-26) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.6e-48:
		tmp = (-1.0 / (B * (1.0 + ((B * B) * (-0.16666666666666666 + ((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984)))))))) - (x / B)
	elif F <= 2.05e-26:
		tmp = 0.0 - (x / B)
	else:
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.6e-48)
		tmp = Float64(Float64(-1.0 / Float64(B * Float64(1.0 + Float64(Float64(B * B) * Float64(-0.16666666666666666 + Float64(Float64(B * B) * Float64(0.008333333333333333 + Float64(Float64(B * B) * -0.0001984126984126984)))))))) - Float64(x / B));
	elseif (F <= 2.05e-26)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 - x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.6e-48)
		tmp = (-1.0 / (B * (1.0 + ((B * B) * (-0.16666666666666666 + ((B * B) * (0.008333333333333333 + ((B * B) * -0.0001984126984126984)))))))) - (x / B);
	elseif (F <= 2.05e-26)
		tmp = 0.0 - (x / B);
	else
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.6e-48], N[(N[(-1.0 / N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(B * B), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(B * B), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.05e-26], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.6000000000000001e-48

   1. Initial program 71.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 37.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 37.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 56.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \color{blue}{\left(B \cdot \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {B}^{2}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({B}^{2} \cdot \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       16. *-lowering-*.f64 37.2%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{5040}\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   11. Simplified 37.2%

       \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right) + -0.16666666666666666\right)\right)}} - \frac{x}{B} \]

   if -5.6000000000000001e-48 < F < 2.0499999999999999e-26

   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 39.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.6%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 2.0499999999999999e-26 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) - x\right), B\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 - x\right)\right), B\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{3} \cdot x\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      11. --lowering--.f64 48.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), B\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(-0.16666666666666666 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 + \left(B \cdot B\right) \cdot -0.0001984126984126984\right)\right)\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 44.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-21}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.4e-48)
   (- (/ -1.0 (* B (+ 1.0 (* (* B B) -0.16666666666666666)))) (/ x B))
   (if (<= F 2.3e-21)
     (- 0.0 (/ x B))
     (/
      (+
       (* (* B B) (+ 0.16666666666666666 (* x 0.3333333333333333)))
       (- 1.0 x))
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.4e-48) {
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B);
	} else if (F <= 2.3e-21) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (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 <= (-9.4d-48)) then
        tmp = ((-1.0d0) / (b * (1.0d0 + ((b * b) * (-0.16666666666666666d0))))) - (x / b)
    else if (f <= 2.3d-21) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = (((b * b) * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (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 <= -9.4e-48) {
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B);
	} else if (F <= 2.3e-21) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.4e-48:
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B)
	elif F <= 2.3e-21:
		tmp = 0.0 - (x / B)
	else:
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.4e-48)
		tmp = Float64(Float64(-1.0 / Float64(B * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666)))) - Float64(x / B));
	elseif (F <= 2.3e-21)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 - x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.4e-48)
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B);
	elseif (F <= 2.3e-21)
		tmp = 0.0 - (x / B);
	else
		tmp = (((B * B) * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 - x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.4e-48], N[(N[(-1.0 / N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-21], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.3999999999999995e-48

   1. Initial program 71.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 37.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 37.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 56.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       6. *-lowering-*.f64 37.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   11. Simplified 37.2%

       \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)}} - \frac{x}{B} \]

   if -9.3999999999999995e-48 < F < 2.29999999999999999e-21

   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 39.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.6%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 2.29999999999999999e-21 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 95.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) - x\right), B\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 - x\right)\right), B\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{3} \cdot x\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

      11. --lowering--.f64 48.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), B\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-21}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 44.6% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.8e-50)
   (- (/ -1.0 (* B (+ 1.0 (* (* B B) -0.16666666666666666)))) (/ x B))
   (if (<= F 5.2e-23)
     (- 0.0 (/ x B))
     (- (+ (/ 1.0 B) (* B 0.16666666666666666)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-50) {
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B);
	} else if (F <= 5.2e-23) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (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 <= (-5.8d-50)) then
        tmp = ((-1.0d0) / (b * (1.0d0 + ((b * b) * (-0.16666666666666666d0))))) - (x / b)
    else if (f <= 5.2d-23) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = ((1.0d0 / b) + (b * 0.16666666666666666d0)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-50) {
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B);
	} else if (F <= 5.2e-23) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.8e-50:
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B)
	elif F <= 5.2e-23:
		tmp = 0.0 - (x / B)
	else:
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.8e-50)
		tmp = Float64(Float64(-1.0 / Float64(B * Float64(1.0 + Float64(Float64(B * B) * -0.16666666666666666)))) - Float64(x / B));
	elseif (F <= 5.2e-23)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * 0.16666666666666666)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.8e-50)
		tmp = (-1.0 / (B * (1.0 + ((B * B) * -0.16666666666666666)))) - (x / B);
	elseif (F <= 5.2e-23)
		tmp = 0.0 - (x / B);
	else
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.8e-50], N[(N[(-1.0 / N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-23], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.80000000000000016e-50

   1. Initial program 71.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 37.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 37.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 56.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{/.f64}\left(x, B\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

       6. *-lowering-*.f64 37.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{/.f64}\left(x, B\right)\right) \]

   11. Simplified 37.2%

       \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)}} - \frac{x}{B} \]

   if -5.80000000000000016e-50 < F < 5.2e-23

   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 39.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.6%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 5.2e-23 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 27.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 27.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\frac{F + \frac{1}{6} \cdot \left({B}^{2} \cdot F\right)}{B}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(F + \frac{1}{6} \cdot \left({B}^{2} \cdot F\right)\right), B\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right)}, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(F + \left(\frac{1}{6} \cdot {B}^{2}\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \color{blue}{2}\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {B}^{2} + 1\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right)}, \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(F, F\right)}, 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right)}, \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {B}^{2}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(F, F\right)}, 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{F}\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{F}\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 20.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   8. Simplified 20.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right) \cdot F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{6} \cdot B + \frac{1}{B}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{B} + \frac{1}{6} \cdot B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{6} \cdot B\right)\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{6} \cdot B\right)\right), \left(\frac{x}{B}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(B \cdot \frac{1}{6}\right)\right), \left(\frac{x}{B}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(B, \frac{1}{6}\right)\right), \left(\frac{x}{B}\right)\right) \]

       7. /-lowering-/.f64 48.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(B, \frac{1}{6}\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   11. Simplified 48.0%

       \[\leadsto \color{blue}{\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot -0.16666666666666666\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 44.5% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-59}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot -0.16666666666666666 + \left(-1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-24}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-59)
   (/ (+ (* (* B B) -0.16666666666666666) (- -1.0 x)) B)
   (if (<= F 1.55e-24)
     (- 0.0 (/ x B))
     (- (+ (/ 1.0 B) (* B 0.16666666666666666)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-59) {
		tmp = (((B * B) * -0.16666666666666666) + (-1.0 - x)) / B;
	} else if (F <= 1.55e-24) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (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 <= (-1.45d-59)) then
        tmp = (((b * b) * (-0.16666666666666666d0)) + ((-1.0d0) - x)) / b
    else if (f <= 1.55d-24) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = ((1.0d0 / b) + (b * 0.16666666666666666d0)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-59) {
		tmp = (((B * B) * -0.16666666666666666) + (-1.0 - x)) / B;
	} else if (F <= 1.55e-24) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-59:
		tmp = (((B * B) * -0.16666666666666666) + (-1.0 - x)) / B
	elif F <= 1.55e-24:
		tmp = 0.0 - (x / B)
	else:
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-59)
		tmp = Float64(Float64(Float64(Float64(B * B) * -0.16666666666666666) + Float64(-1.0 - x)) / B);
	elseif (F <= 1.55e-24)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * 0.16666666666666666)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45e-59)
		tmp = (((B * B) * -0.16666666666666666) + (-1.0 - x)) / B;
	elseif (F <= 1.55e-24)
		tmp = 0.0 - (x / B);
	else
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-59], N[(N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.55e-24], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.45000000000000008e-59

   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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 38.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 38.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{\color{blue}{x}}{B} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 56.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot {B}^{2} - \left(1 + x\right)}{B}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{6} \cdot {B}^{2} - \left(1 + x\right)\right), \color{blue}{B}\right) \]

       2. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {B}^{2} - 1\right) - x\right), B\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {B}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right) - x\right), B\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {B}^{2} + -1\right) - x\right), B\right) \]

       5. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{6} \cdot {B}^{2} + \left(-1 - x\right)\right), B\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot {B}^{2}\right), \left(-1 - x\right)\right), B\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot \frac{-1}{6}\right), \left(-1 - x\right)\right), B\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right), \left(-1 - x\right)\right), B\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right), \left(-1 - x\right)\right), B\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right), \left(-1 - x\right)\right), B\right) \]

       11. --lowering--.f64 36.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right), \mathsf{\_.f64}\left(-1, x\right)\right), B\right) \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot -0.16666666666666666 + \left(-1 - x\right)}{B}} \]

   if -1.45000000000000008e-59 < F < 1.55e-24

   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 39.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.8%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 1.55e-24 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 27.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 27.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\frac{F + \frac{1}{6} \cdot \left({B}^{2} \cdot F\right)}{B}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(F + \frac{1}{6} \cdot \left({B}^{2} \cdot F\right)\right), B\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right)}, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(F + \left(\frac{1}{6} \cdot {B}^{2}\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \color{blue}{2}\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {B}^{2} + 1\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right)}, \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(F, F\right)}, 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right)}, \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {B}^{2}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(F, F\right)}, 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{F}\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{F}\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 20.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   8. Simplified 20.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right) \cdot F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{6} \cdot B + \frac{1}{B}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{B} + \frac{1}{6} \cdot B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{6} \cdot B\right)\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{6} \cdot B\right)\right), \left(\frac{x}{B}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(B \cdot \frac{1}{6}\right)\right), \left(\frac{x}{B}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(B, \frac{1}{6}\right)\right), \left(\frac{x}{B}\right)\right) \]

       7. /-lowering-/.f64 48.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(B, \frac{1}{6}\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   11. Simplified 48.0%

       \[\leadsto \color{blue}{\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-59}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot -0.16666666666666666 + \left(-1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-24}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 44.5% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.14 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.14e-54)
   (/ (- -1.0 x) B)
   (if (<= F 1.6e-22)
     (- 0.0 (/ x B))
     (- (+ (/ 1.0 B) (* B 0.16666666666666666)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.14e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.6e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (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 <= (-1.14d-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.6d-22) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = ((1.0d0 / b) + (b * 0.16666666666666666d0)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.14e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.6e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.14e-54:
		tmp = (-1.0 - x) / B
	elif F <= 1.6e-22:
		tmp = 0.0 - (x / B)
	else:
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.14e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.6e-22)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * 0.16666666666666666)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.14e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.6e-22)
		tmp = 0.0 - (x / B);
	else
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.14e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.6e-22], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.14000000000000004e-54

   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 30.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 30.8%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   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 36.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.14000000000000004e-54 < F < 1.59999999999999994e-22

   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 39.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.8%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 1.59999999999999994e-22 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 27.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 27.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\frac{F + \frac{1}{6} \cdot \left({B}^{2} \cdot F\right)}{B}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(F + \frac{1}{6} \cdot \left({B}^{2} \cdot F\right)\right), B\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right)}, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(F + \left(\frac{1}{6} \cdot {B}^{2}\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \color{blue}{2}\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {B}^{2} + 1\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right)}, \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right) \cdot F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(F, F\right)}, 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right)}, \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {B}^{2}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(F, F\right)}, 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{F}\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{F}\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 20.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{1}{6}\right)\right), F\right), B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   8. Simplified 20.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right) \cdot F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{6} \cdot B + \frac{1}{B}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{B} + \frac{1}{6} \cdot B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{6} \cdot B\right)\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{6} \cdot B\right)\right), \left(\frac{x}{B}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(B \cdot \frac{1}{6}\right)\right), \left(\frac{x}{B}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(B, \frac{1}{6}\right)\right), \left(\frac{x}{B}\right)\right) \]

       7. /-lowering-/.f64 48.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{*.f64}\left(B, \frac{1}{6}\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   11. Simplified 48.0%

       \[\leadsto \color{blue}{\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.14 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 44.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-52)
   (/ (- -1.0 x) B)
   (if (<= F 1.4e-22) (- 0.0 (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-52) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-22) {
		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 <= (-9d-52)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.4d-22) 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 <= -9e-52) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9e-52:
		tmp = (-1.0 - x) / B
	elif F <= 1.4e-22:
		tmp = 0.0 - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-52)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.4e-22)
		tmp = Float64(0.0 - Float64(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 <= -9e-52)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.4e-22)
		tmp = 0.0 - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-52], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.4e-22], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.0000000000000001e-52

   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 30.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 30.8%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   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 36.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -9.0000000000000001e-52 < F < 1.39999999999999997e-22

   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 39.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 26.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 26.8%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]

   if 1.39999999999999997e-22 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 43.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   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 47.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 37.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-51) (/ (- -1.0 x) B) (- 0.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-51) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 0.0 - (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 <= (-1.6d-51)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = 0.0d0 - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-51) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 0.0 - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-51:
		tmp = (-1.0 - x) / B
	else:
		tmp = 0.0 - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-51)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(0.0 - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.6e-51)
		tmp = (-1.0 - x) / B;
	else
		tmp = 0.0 - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-51], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.6e-51

   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 30.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 30.8%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   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 36.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.6e-51 < F

   1. Initial program 83.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x\right)}, B\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x\right)\right), B\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      3. --lowering--.f64 29.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 29.2%

      \[\leadsto \frac{\color{blue}{0 - x}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 30.3% accurate, 64.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1 - x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- -1.0 x) B))

C

double code(double F, double B, double x) {
	return (-1.0 - x) / 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) - x) / b
end function

Java

public static double code(double F, double B, double x) {
	return (-1.0 - x) / B;
}

Python

def code(F, B, x):
	return (-1.0 - x) / B

Julia

function code(F, B, x)
	return Float64(Float64(-1.0 - x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = (-1.0 - x) / B;
end

Wolfram

code[F_, B_, x_] := N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 37.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

5. Simplified 37.4%

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

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 26.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

8. Simplified 26.6%

   \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]
9. Add Preprocessing

Alternative 35: 10.9% 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 79.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 37.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

5. Simplified 37.4%

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

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 26.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

8. Simplified 26.6%

   \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 9.3%

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{B}\right) \]

11. Simplified 9.3%

    \[\leadsto \color{blue}{\frac{-1}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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))))))