VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.4%

Time: 21.4s

Alternatives: 29

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.32 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.32e+17)
     (- (/ -1.0 (sin B)) (* (/ x (sin B)) (cos B)))
     (if (<= F 0.14)
       (-
        (* (/ 1.0 (/ (sin B) F)) (pow (+ (* F F) (- 2.0 (* x -2.0))) -0.5))
        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 <= -1.32e+17) {
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	} else if (F <= 0.14) {
		tmp = ((1.0 / (sin(B) / F)) * pow(((F * F) + (2.0 - (x * -2.0))), -0.5)) - 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 <= (-1.32d+17)) then
        tmp = ((-1.0d0) / sin(b)) - ((x / sin(b)) * cos(b))
    else if (f <= 0.14d0) then
        tmp = ((1.0d0 / (sin(b) / f)) * (((f * f) + (2.0d0 - (x * (-2.0d0)))) ** (-0.5d0))) - 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 <= -1.32e+17) {
		tmp = (-1.0 / Math.sin(B)) - ((x / Math.sin(B)) * Math.cos(B));
	} else if (F <= 0.14) {
		tmp = ((1.0 / (Math.sin(B) / F)) * Math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5)) - 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 <= -1.32e+17:
		tmp = (-1.0 / math.sin(B)) - ((x / math.sin(B)) * math.cos(B))
	elif F <= 0.14:
		tmp = ((1.0 / (math.sin(B) / F)) * math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5)) - 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 <= -1.32e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	elseif (F <= 0.14)
		tmp = Float64(Float64(Float64(1.0 / Float64(sin(B) / F)) * (Float64(Float64(F * F) + Float64(2.0 - Float64(x * -2.0))) ^ -0.5)) - 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 <= -1.32e+17)
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	elseif (F <= 0.14)
		tmp = ((1.0 / (sin(B) / F)) * (((F * F) + (2.0 - (x * -2.0))) ^ -0.5)) - 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, -1.32e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -1.32e17

   1. Initial program 65.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.7%

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

   5. Simplified 99.7%

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

      1. div-inv N/A

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

      2. tan-quot N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. cos-lowering-cos.f64 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -1.32e17 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. clear-num N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.32 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.35e+17)
     (- (/ -1.0 (sin B)) (* (/ x (sin B)) (cos B)))
     (if (<= F 0.14)
       (- (* F (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B))) 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 <= -1.35e+17) {
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	} else if (F <= 0.14) {
		tmp = (F * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - 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 <= (-1.35d+17)) then
        tmp = ((-1.0d0) / sin(b)) - ((x / sin(b)) * cos(b))
    else if (f <= 0.14d0) then
        tmp = (f * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - 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 <= -1.35e+17) {
		tmp = (-1.0 / Math.sin(B)) - ((x / Math.sin(B)) * Math.cos(B));
	} else if (F <= 0.14) {
		tmp = (F * (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - 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 <= -1.35e+17:
		tmp = (-1.0 / math.sin(B)) - ((x / math.sin(B)) * math.cos(B))
	elif F <= 0.14:
		tmp = (F * (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - 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 <= -1.35e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	elseif (F <= 0.14)
		tmp = Float64(Float64(F * Float64((Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5) / sin(B))) - 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 <= -1.35e+17)
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	elseif (F <= 0.14)
		tmp = (F * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - 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, -1.35e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], 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] - 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 < -1.35e17

   1. Initial program 65.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.7%

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

   5. Simplified 99.7%

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

      1. div-inv N/A

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

      2. tan-quot N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. cos-lowering-cos.f64 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -1.35e17 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+131}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+131)
     (- (/ -1.0 (sin B)) (* (/ x (sin B)) (cos B)))
     (if (<= F 0.14)
       (- (/ (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) (sin B)) 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 <= -4e+131) {
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	} else if (F <= 0.14) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) / sin(B)) - 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 <= (-4d+131)) then
        tmp = ((-1.0d0) / sin(b)) - ((x / sin(b)) * cos(b))
    else if (f <= 0.14d0) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) / sin(b)) - 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 <= -4e+131) {
		tmp = (-1.0 / Math.sin(B)) - ((x / Math.sin(B)) * Math.cos(B));
	} else if (F <= 0.14) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) / Math.sin(B)) - 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 <= -4e+131:
		tmp = (-1.0 / math.sin(B)) - ((x / math.sin(B)) * math.cos(B))
	elif F <= 0.14:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) / math.sin(B)) - 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 <= -4e+131)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	elseif (F <= 0.14)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) / sin(B)) - 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 <= -4e+131)
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	elseif (F <= 0.14)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) / sin(B)) - 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, -4e+131], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(N[(F / N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -3.9999999999999996e131

   1. Initial program 47.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. div-inv N/A

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

      2. tan-quot N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. cos-lowering-cos.f64 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -3.9999999999999996e131 < F < 0.14000000000000001

   1. Initial program 99.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 99.6%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+131}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.95:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.95)
     (-
      (/ (/ F (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))) (sin B))
      t_0)
     (if (<= F 0.14)
       (- (/ (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (sin B)) 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 <= -0.95) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) / sin(B)) - 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 <= (-0.95d0)) then
        tmp = ((f / (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f))))) / sin(b)) - t_0
    else if (f <= 0.14d0) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) / sin(b)) - 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 <= -0.95) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / Math.sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) / Math.sin(B)) - 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 <= -0.95:
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / math.sin(B)) - t_0
	elif F <= 0.14:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) / math.sin(B)) - 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 <= -0.95)
		tmp = Float64(Float64(Float64(F / Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))))) / sin(B)) - t_0);
	elseif (F <= 0.14)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) / sin(B)) - 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 <= -0.95)
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	elseif (F <= 0.14)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) / sin(B)) - 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, -0.95], N[(N[(N[(F / 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] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -0.94999999999999996

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 66.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in F around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

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

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

   11. Simplified 99.2%

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

   if -0.94999999999999996 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. distribute-lft-neg-in N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

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

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

      14. sin-lowering-sin.f64 99.3%

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

   7. Simplified 99.3%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.95:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.9:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.9)
     (-
      (/ (/ F (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))) (sin B))
      t_0)
     (if (<= F 0.14)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) 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 <= -0.9) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - 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 <= (-0.9d0)) then
        tmp = ((f / (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f))))) / sin(b)) - t_0
    else if (f <= 0.14d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - 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 <= -0.9) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / Math.sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - 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 <= -0.9:
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / math.sin(B)) - t_0
	elif F <= 0.14:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - 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 <= -0.9)
		tmp = Float64(Float64(Float64(F / Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))))) / sin(B)) - t_0);
	elseif (F <= 0.14)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - 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 <= -0.9)
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	elseif (F <= 0.14)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - 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, -0.9], N[(N[(N[(F / 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] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $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 < -0.900000000000000022

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 66.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in F around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

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

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

   11. Simplified 99.2%

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

   if -0.900000000000000022 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 99.3%

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

   9. Simplified 99.3%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.9:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.9:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + x \cdot 2}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.9)
     (-
      (/ (/ F (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))) (sin B))
      t_0)
     (if (<= F 0.14)
       (- (/ (/ F (sqrt (+ 2.0 (* x 2.0)))) (sin B)) 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 <= -0.9) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = ((F / sqrt((2.0 + (x * 2.0)))) / sin(B)) - 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 <= (-0.9d0)) then
        tmp = ((f / (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f))))) / sin(b)) - t_0
    else if (f <= 0.14d0) then
        tmp = ((f / sqrt((2.0d0 + (x * 2.0d0)))) / sin(b)) - 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 <= -0.9) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / Math.sin(B)) - t_0;
	} else if (F <= 0.14) {
		tmp = ((F / Math.sqrt((2.0 + (x * 2.0)))) / Math.sin(B)) - 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 <= -0.9:
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / math.sin(B)) - t_0
	elif F <= 0.14:
		tmp = ((F / math.sqrt((2.0 + (x * 2.0)))) / math.sin(B)) - 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 <= -0.9)
		tmp = Float64(Float64(Float64(F / Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))))) / sin(B)) - t_0);
	elseif (F <= 0.14)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(x * 2.0)))) / sin(B)) - 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 <= -0.9)
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	elseif (F <= 0.14)
		tmp = ((F / sqrt((2.0 + (x * 2.0)))) / sin(B)) - 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, -0.9], N[(N[(N[(F / 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] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.14], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -0.900000000000000022

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 66.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in F around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

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

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

   11. Simplified 99.2%

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

   if -0.900000000000000022 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in F around 0

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

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

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

       2. metadata-eval N/A

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

       3. distribute-lft-neg-in N/A

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

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

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

       5. distribute-lft-neg-in N/A

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

       6. metadata-eval N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 99.3%

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

   11. Simplified 99.3%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.9:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e-5)
     (-
      (/ (/ F (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))) (sin B))
      t_0)
     (if (<= F -4e-147)
       (- (* F (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B))) (/ x B))
       (if (<= F 0.14)
         (-
          (*
           (pow (+ (* F F) (- 2.0 (* x -2.0))) -0.5)
           (/ F (* B (+ 1.0 (* B (* B -0.16666666666666666))))))
          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 <= -1.7e-5) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	} else if (F <= -4e-147) {
		tmp = (F * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - 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 <= (-1.7d-5)) then
        tmp = ((f / (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f))))) / sin(b)) - t_0
    else if (f <= (-4d-147)) then
        tmp = (f * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = ((((f * f) + (2.0d0 - (x * (-2.0d0)))) ** (-0.5d0)) * (f / (b * (1.0d0 + (b * (b * (-0.16666666666666666d0))))))) - 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 <= -1.7e-5) {
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / Math.sin(B)) - t_0;
	} else if (F <= -4e-147) {
		tmp = (F * (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (Math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - 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 <= -1.7e-5:
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / math.sin(B)) - t_0
	elif F <= -4e-147:
		tmp = (F * (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - (x / B)
	elif F <= 0.14:
		tmp = (math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - 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 <= -1.7e-5)
		tmp = Float64(Float64(Float64(F / Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))))) / sin(B)) - t_0);
	elseif (F <= -4e-147)
		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((Float64(Float64(F * F) + Float64(2.0 - Float64(x * -2.0))) ^ -0.5) * Float64(F / Float64(B * Float64(1.0 + Float64(B * Float64(B * -0.16666666666666666)))))) - 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 <= -1.7e-5)
		tmp = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	elseif (F <= -4e-147)
		tmp = (F * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - (x / B);
	elseif (F <= 0.14)
		tmp = ((((F * F) + (2.0 - (x * -2.0))) ^ -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - 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, -1.7e-5], N[(N[(N[(F / 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] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4e-147], 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[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[(B * N[(1.0 + N[(B * N[(B * -0.16666666666666666), $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 4 regimes

2. if F < -1.7e-5

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 67.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 79.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 79.1%

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

   9. Taylor expanded in F around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

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

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

   11. Simplified 99.2%

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

   if -1.7e-5 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 79.3%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 85.9%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* F (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B)))
          (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -2.15e-5)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4e-151)
       t_0
       (if (<= F 1.25e-143)
         (/ x (- 0.0 (tan B)))
         (if (<= F 0.029) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.15e-5) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4e-151) {
		tmp = t_0;
	} else if (F <= 1.25e-143) {
		tmp = x / (0.0 - tan(B));
	} else if (F <= 0.029) {
		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 * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-2.15d-5)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4d-151)) then
        tmp = t_0
    else if (f <= 1.25d-143) then
        tmp = x / (0.0d0 - tan(b))
    else if (f <= 0.029d0) 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.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.15e-5) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4e-151) {
		tmp = t_0;
	} else if (F <= 1.25e-143) {
		tmp = x / (0.0 - Math.tan(B));
	} else if (F <= 0.029) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F * (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.15e-5:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4e-151:
		tmp = t_0
	elif F <= 1.25e-143:
		tmp = x / (0.0 - math.tan(B))
	elif F <= 0.029:
		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((Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5) / sin(B))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.15e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4e-151)
		tmp = t_0;
	elseif (F <= 1.25e-143)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	elseif (F <= 0.029)
		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 * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.15e-5)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4e-151)
		tmp = t_0;
	elseif (F <= 1.25e-143)
		tmp = x / (0.0 - tan(B));
	elseif (F <= 0.029)
		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[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]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.15e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4e-151], t$95$0, If[LessEqual[F, 1.25e-143], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.029], 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 < -2.1500000000000001e-5

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 67.3%

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

   5. Taylor expanded in F around -inf

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

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

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

      2. sin-lowering-sin.f64 98.7%

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

   7. Simplified 98.7%

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

   if -2.1500000000000001e-5 < F < -3.9999999999999998e-151 or 1.2500000000000001e-143 < F < 0.0290000000000000015

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 78.3%

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

   if -3.9999999999999998e-151 < F < 1.2500000000000001e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   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. 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-lowering-neg.f64 N/A

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

      7. sin-lowering-sin.f64 88.3%

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

   9. Simplified 88.3%

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

       1. distribute-frac-neg2 N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

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

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

       8. tan-lowering-tan.f64 88.4%

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

   11. Applied egg-rr 88.4%

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

   if 0.0290000000000000015 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-151}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}}}{\sin B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.000104:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.096:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (- (/ (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) (sin B)) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -0.000104)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.5e-151)
       t_0
       (if (<= F 2.2e-145)
         (/ x (- 0.0 (tan B)))
         (if (<= F 0.096) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) / sin(B)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.000104) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.5e-151) {
		tmp = t_0;
	} else if (F <= 2.2e-145) {
		tmp = x / (0.0 - tan(B));
	} else if (F <= 0.096) {
		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 / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) / sin(b)) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-0.000104d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.5d-151)) then
        tmp = t_0
    else if (f <= 2.2d-145) then
        tmp = x / (0.0d0 - tan(b))
    else if (f <= 0.096d0) 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.sqrt(((F * F) + (2.0 + (x * 2.0))))) / Math.sin(B)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.000104) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.5e-151) {
		tmp = t_0;
	} else if (F <= 2.2e-145) {
		tmp = x / (0.0 - Math.tan(B));
	} else if (F <= 0.096) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) / math.sin(B)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.000104:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.5e-151:
		tmp = t_0
	elif F <= 2.2e-145:
		tmp = x / (0.0 - math.tan(B))
	elif F <= 0.096:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) / sin(B)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.000104)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.5e-151)
		tmp = t_0;
	elseif (F <= 2.2e-145)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	elseif (F <= 0.096)
		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 / sqrt(((F * F) + (2.0 + (x * 2.0))))) / sin(B)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.000104)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.5e-151)
		tmp = t_0;
	elseif (F <= 2.2e-145)
		tmp = x / (0.0 - tan(B));
	elseif (F <= 0.096)
		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[(N[(F / N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.000104], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.5e-151], t$95$0, If[LessEqual[F, 2.2e-145], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.096], 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 < -1.03999999999999994e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 67.3%

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

   5. Taylor expanded in F around -inf

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

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

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

      2. sin-lowering-sin.f64 98.7%

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

   7. Simplified 98.7%

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

   if -1.03999999999999994e-4 < F < -3.49999999999999995e-151 or 2.19999999999999999e-145 < F < 0.096000000000000002

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in B around 0

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

       1. /-lowering-/.f64 78.3%

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

   11. Simplified 78.3%

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

   if -3.49999999999999995e-151 < F < 2.19999999999999999e-145

   1. Initial program 99.3%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   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. 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-lowering-neg.f64 N/A

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

      7. sin-lowering-sin.f64 88.3%

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

   9. Simplified 88.3%

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

       1. distribute-frac-neg2 N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

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

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

       8. tan-lowering-tan.f64 88.4%

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

   11. Applied egg-rr 88.4%

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

   if 0.096000000000000002 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.000104:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.096:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-147}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00049)
     (-
      (/ (/ F (* F (+ -1.0 (/ (- -1.0 (* (* x 2.0) 0.5)) (* F F))))) (sin B))
      t_0)
     (if (<= F -3.5e-147)
       (- (* F (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B))) (/ x B))
       (if (<= F 0.14)
         (- (* (pow (+ (* F F) (- 2.0 (* x -2.0))) -0.5) (/ F B)) 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 = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	} else if (F <= -3.5e-147) {
		tmp = (F * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / B)) - 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 = ((f / (f * ((-1.0d0) + (((-1.0d0) - ((x * 2.0d0) * 0.5d0)) / (f * f))))) / sin(b)) - t_0
    else if (f <= (-3.5d-147)) then
        tmp = (f * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = ((((f * f) + (2.0d0 - (x * (-2.0d0)))) ** (-0.5d0)) * (f / b)) - 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 = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / Math.sin(B)) - t_0;
	} else if (F <= -3.5e-147) {
		tmp = (F * (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (Math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / B)) - 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 = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / math.sin(B)) - t_0
	elif F <= -3.5e-147:
		tmp = (F * (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - (x / B)
	elif F <= 0.14:
		tmp = (math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / B)) - 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(Float64(F / Float64(F * Float64(-1.0 + Float64(Float64(-1.0 - Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))))) / sin(B)) - t_0);
	elseif (F <= -3.5e-147)
		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((Float64(Float64(F * F) + Float64(2.0 - Float64(x * -2.0))) ^ -0.5) * Float64(F / B)) - 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 = ((F / (F * (-1.0 + ((-1.0 - ((x * 2.0) * 0.5)) / (F * F))))) / sin(B)) - t_0;
	elseif (F <= -3.5e-147)
		tmp = (F * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - (x / B);
	elseif (F <= 0.14)
		tmp = ((((F * F) + (2.0 - (x * -2.0))) ^ -0.5) * (F / B)) - 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[(N[(F / 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] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.5e-147], 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[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 67.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 79.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 79.1%

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

   9. Taylor expanded in F around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

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

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

   11. Simplified 99.2%

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

   if -4.8999999999999998e-4 < F < -3.50000000000000004e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 79.3%

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

   if -3.50000000000000004e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 85.5%

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

   7. Simplified 85.5%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{\frac{F}{F \cdot \left(-1 + \frac{-1 - \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-147}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

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 (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (sin B))) (/ x B))
       (if (<= F 0.14)
         (- (* (pow (+ (* F F) (- 2.0 (* x -2.0))) -0.5) (/ F B)) 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 * (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / B)) - 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 * ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / sin(b))) - (x / b)
    else if (f <= 0.14d0) then
        tmp = ((((f * f) + (2.0d0 - (x * (-2.0d0)))) ** (-0.5d0)) * (f / b)) - 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.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / Math.sin(B))) - (x / B);
	} else if (F <= 0.14) {
		tmp = (Math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / B)) - 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.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / math.sin(B))) - (x / B)
	elif F <= 0.14:
		tmp = (math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / B)) - 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(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((Float64(Float64(F * F) + Float64(2.0 - Float64(x * -2.0))) ^ -0.5) * Float64(F / B)) - 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 * ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / sin(B))) - (x / B);
	elseif (F <= 0.14)
		tmp = ((((F * F) + (2.0 - (x * -2.0))) ^ -0.5) * (F / B)) - 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[(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[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 67.3%

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

   5. Taylor expanded in F around -inf

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

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

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

      2. sin-lowering-sin.f64 98.7%

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

   7. 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 79.3%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 85.5%

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

   7. Simplified 85.5%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00049:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-147}:\\ \;\;\;\;F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{F \cdot F + 2}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

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 -1.8e-5)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.3e-114)
       t_0
       (if (<= F 2.6e-135)
         (/ x (- 0.0 (tan B)))
         (if (<= F 0.14) 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 <= -1.8e-5) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.3e-114) {
		tmp = t_0;
	} else if (F <= 2.6e-135) {
		tmp = x / (0.0 - tan(B));
	} else if (F <= 0.14) {
		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 <= (-1.8d-5)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.3d-114)) then
        tmp = t_0
    else if (f <= 2.6d-135) then
        tmp = x / (0.0d0 - tan(b))
    else if (f <= 0.14d0) 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 <= -1.8e-5) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.3e-114) {
		tmp = t_0;
	} else if (F <= 2.6e-135) {
		tmp = x / (0.0 - Math.tan(B));
	} else if (F <= 0.14) {
		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 <= -1.8e-5:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.3e-114:
		tmp = t_0
	elif F <= 2.6e-135:
		tmp = x / (0.0 - math.tan(B))
	elif F <= 0.14:
		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 <= -1.8e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.3e-114)
		tmp = t_0;
	elseif (F <= 2.6e-135)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	elseif (F <= 0.14)
		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 <= -1.8e-5)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.3e-114)
		tmp = t_0;
	elseif (F <= 2.6e-135)
		tmp = x / (0.0 - tan(B));
	elseif (F <= 0.14)
		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, -1.8e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.3e-114], t$95$0, If[LessEqual[F, 2.6e-135], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.14], 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 < -1.80000000000000005e-5

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 67.3%

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

   5. Taylor expanded in F around -inf

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

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

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

      2. sin-lowering-sin.f64 98.7%

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

   7. Simplified 98.7%

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

   if -1.80000000000000005e-5 < F < -3.30000000000000035e-114 or 2.60000000000000004e-135 < 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. 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 -3.30000000000000035e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   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. 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-lowering-neg.f64 N/A

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

      7. sin-lowering-sin.f64 83.6%

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

   9. Simplified 83.6%

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

       1. distribute-frac-neg2 N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

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

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

       8. tan-lowering-tan.f64 83.7%

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

   11. Applied egg-rr 83.7%

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

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 53.1%

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

   5. Taylor expanded in F around inf

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

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

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

      2. sin-lowering-sin.f64 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{F \cdot F + 2}}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.8e-29)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.1e-92)
       (/ x (- 0.0 (tan B)))
       (if (<= F 9.5e-7)
         (-
          (*
           (pow (+ (* F F) (- 2.0 (* x -2.0))) -0.5)
           (/ F (* B (+ 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 <= -8.8e-29) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.1e-92) {
		tmp = x / (0.0 - tan(B));
	} else if (F <= 9.5e-7) {
		tmp = (pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (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 <= (-8.8d-29)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.1d-92) then
        tmp = x / (0.0d0 - tan(b))
    else if (f <= 9.5d-7) then
        tmp = ((((f * f) + (2.0d0 - (x * (-2.0d0)))) ** (-0.5d0)) * (f / (b * (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 <= -8.8e-29) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.1e-92) {
		tmp = x / (0.0 - Math.tan(B));
	} else if (F <= 9.5e-7) {
		tmp = (Math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (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 <= -8.8e-29:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.1e-92:
		tmp = x / (0.0 - math.tan(B))
	elif F <= 9.5e-7:
		tmp = (math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (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 <= -8.8e-29)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.1e-92)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	elseif (F <= 9.5e-7)
		tmp = Float64(Float64((Float64(Float64(F * F) + Float64(2.0 - Float64(x * -2.0))) ^ -0.5) * Float64(F / Float64(B * Float64(1.0 + Float64(B * Float64(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 <= -8.8e-29)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.1e-92)
		tmp = x / (0.0 - tan(B));
	elseif (F <= 9.5e-7)
		tmp = ((((F * F) + (2.0 - (x * -2.0))) ^ -0.5) * (F / (B * (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, -8.8e-29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.1e-92], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-7], N[(N[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[(B * N[(1.0 + N[(B * N[(B * -0.16666666666666666), $MachinePrecision]), $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 < -8.79999999999999961e-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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   3. Simplified 70.0%

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

   5. Taylor expanded in F around -inf

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

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

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

      2. sin-lowering-sin.f64 92.4%

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

   7. Simplified 92.4%

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

   if -8.79999999999999961e-29 < F < 1.09999999999999994e-92

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 72.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

   if 1.09999999999999994e-92 < F < 9.5000000000000001e-7

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 73.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 73.2%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 56.9%

       \[\leadsto \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\color{blue}{B}} \]

   if 9.5000000000000001e-7 < 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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.0088:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.5e-29)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 6e-93)
       (/ x (- 0.0 (tan B)))
       (if (<= F 0.0088)
         (-
          (*
           (pow (+ (* F F) (- 2.0 (* x -2.0))) -0.5)
           (/ F (* B (+ 1.0 (* B (* B -0.16666666666666666))))))
          (/ x B))
         (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.5e-29) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 6e-93) {
		tmp = x / (0.0 - tan(B));
	} else if (F <= 0.0088) {
		tmp = (pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-2.5d-29)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 6d-93) then
        tmp = x / (0.0d0 - tan(b))
    else if (f <= 0.0088d0) then
        tmp = ((((f * f) + (2.0d0 - (x * (-2.0d0)))) ** (-0.5d0)) * (f / (b * (1.0d0 + (b * (b * (-0.16666666666666666d0))))))) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -2.5e-29) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 6e-93) {
		tmp = x / (0.0 - Math.tan(B));
	} else if (F <= 0.0088) {
		tmp = (Math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.5e-29:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 6e-93:
		tmp = x / (0.0 - math.tan(B))
	elif F <= 0.0088:
		tmp = (math.pow(((F * F) + (2.0 - (x * -2.0))), -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.5e-29)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 6e-93)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	elseif (F <= 0.0088)
		tmp = Float64(Float64((Float64(Float64(F * F) + Float64(2.0 - Float64(x * -2.0))) ^ -0.5) * Float64(F / Float64(B * Float64(1.0 + Float64(B * Float64(B * -0.16666666666666666)))))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / 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 <= -2.5e-29)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 6e-93)
		tmp = x / (0.0 - tan(B));
	elseif (F <= 0.0088)
		tmp = ((((F * F) + (2.0 - (x * -2.0))) ^ -0.5) * (F / (B * (1.0 + (B * (B * -0.16666666666666666)))))) - (x / B);
	else
		tmp = (1.0 / 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, -2.5e-29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6e-93], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0088], N[(N[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[(B * N[(1.0 + N[(B * N[(B * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.49999999999999993e-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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 92.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -2.49999999999999993e-29 < F < 6.0000000000000003e-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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 72.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

   if 6.0000000000000003e-93 < F < 0.00880000000000000053

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 73.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 73.2%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 56.9%

       \[\leadsto \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\color{blue}{B}} \]

   if 0.00880000000000000053 < 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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 99.7%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 78.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Simplified 78.3%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{elif}\;F \leq 0.0088:\\ \;\;\;\;{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} \cdot \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.014:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F \cdot \left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right)\right) + 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.014)
   (/
    (-
     (+
      (*
       (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
       (* F (+ 1.0 (* (* B B) 0.16666666666666666))))
      (* 0.3333333333333333 (* x (* B B))))
     x)
    B)
   (/ x (- 0.0 (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.014) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + (0.3333333333333333 * (x * (B * B)))) - x) / B;
	} else {
		tmp = x / (0.0 - tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.014d0) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f * (1.0d0 + ((b * b) * 0.16666666666666666d0)))) + (0.3333333333333333d0 * (x * (b * b)))) - x) / b
    else
        tmp = x / (0.0d0 - tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.014) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + (0.3333333333333333 * (x * (B * B)))) - x) / B;
	} else {
		tmp = x / (0.0 - Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.014:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + (0.3333333333333333 * (x * (B * B)))) - x) / B
	else:
		tmp = x / (0.0 - math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.014)
		tmp = Float64(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(0.3333333333333333 * Float64(x * Float64(B * B)))) - x) / B);
	else
		tmp = Float64(x / Float64(0.0 - tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.014)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F * (1.0 + ((B * B) * 0.16666666666666666)))) + (0.3333333333333333 * (x * (B * B)))) - x) / B;
	else
		tmp = x / (0.0 - tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.014], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F * N[(1.0 + N[(N[(B * B), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0140000000000000003

   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%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot \left(\left(\left(B \cdot B\right) \cdot 0.16666666666666666 + 1\right) \cdot F\right) + 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right)\right) - x}{B}} \]

   if 0.0140000000000000003 < 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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 63.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 64.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 64.1%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.014:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F \cdot \left(1 + \left(B \cdot B\right) \cdot 0.16666666666666666\right)\right) + 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{-1}{B}}{1 + -0.16666666666666666 \cdot \left(B \cdot B\right)} - t\_0\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.6e+131)
     (- (/ (/ -1.0 B) (+ 1.0 (* -0.16666666666666666 (* B B)))) t_0)
     (if (<= F -2.25e+17)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 4.4e-30) (/ x (- 0.0 (tan B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.6e+131) {
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - t_0;
	} else if (F <= -2.25e+17) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.4e-30) {
		tmp = x / (0.0 - tan(B));
	} else {
		tmp = (1.0 / 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 <= (-3.6d+131)) then
        tmp = (((-1.0d0) / b) / (1.0d0 + ((-0.16666666666666666d0) * (b * b)))) - t_0
    else if (f <= (-2.25d+17)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.4d-30) then
        tmp = x / (0.0d0 - tan(b))
    else
        tmp = (1.0d0 / 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 <= -3.6e+131) {
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - t_0;
	} else if (F <= -2.25e+17) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.4e-30) {
		tmp = x / (0.0 - Math.tan(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.6e+131:
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - t_0
	elif F <= -2.25e+17:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.4e-30:
		tmp = x / (0.0 - math.tan(B))
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.6e+131)
		tmp = Float64(Float64(Float64(-1.0 / B) / Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B)))) - t_0);
	elseif (F <= -2.25e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.4e-30)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	else
		tmp = Float64(Float64(1.0 / 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 <= -3.6e+131)
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - t_0;
	elseif (F <= -2.25e+17)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.4e-30)
		tmp = x / (0.0 - tan(B));
	else
		tmp = (1.0 / 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, -3.6e+131], N[(N[(N[(-1.0 / B), $MachinePrecision] / N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.25e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.4e-30], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.60000000000000031e131

   1. Initial program 47.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 47.2%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(1\right)}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{B}\right)}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666}} - \frac{x}{\tan B} \]

   if -3.60000000000000031e131 < F < -2.25e17

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 75.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   8. Simplified 75.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -2.25e17 < F < 4.39999999999999967e-30

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 65.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 65.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

   if 4.39999999999999967e-30 < F

   1. Initial program 56.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 93.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 74.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{-1}{B}}{1 + -0.16666666666666666 \cdot \left(B \cdot B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+131}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e+131)
     (- (/ -1.0 B) t_0)
     (if (<= F -2.25e+17)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 3.8e-30) (/ x (- 0.0 (tan B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.7e+131) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -2.25e+17) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.8e-30) {
		tmp = x / (0.0 - tan(B));
	} else {
		tmp = (1.0 / 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 <= (-1.7d+131)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-2.25d+17)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.8d-30) then
        tmp = x / (0.0d0 - tan(b))
    else
        tmp = (1.0d0 / 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 <= -1.7e+131) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -2.25e+17) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.8e-30) {
		tmp = x / (0.0 - Math.tan(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.7e+131:
		tmp = (-1.0 / B) - t_0
	elif F <= -2.25e+17:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.8e-30:
		tmp = x / (0.0 - math.tan(B))
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.7e+131)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -2.25e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.8e-30)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	else
		tmp = Float64(Float64(1.0 / 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 <= -1.7e+131)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -2.25e+17)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.8e-30)
		tmp = x / (0.0 - tan(B));
	else
		tmp = (1.0 / 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, -1.7e+131], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.25e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e-30], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.69999999999999993e131

   1. Initial program 47.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 47.2%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(1\right)}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{B}\right)}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 84.3%

       \[\leadsto \color{blue}{\frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 83.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 83.9%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -1.69999999999999993e131 < F < -2.25e17

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 75.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   8. Simplified 75.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -2.25e17 < F < 3.8000000000000003e-30

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 65.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 65.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

   if 3.8000000000000003e-30 < F

   1. Initial program 56.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 93.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 74.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+131}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 5e-5)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (/ x (- 0.0 (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 5e-5) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = x / (0.0 - tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 5d-5) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = x / (0.0d0 - tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 5e-5) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = x / (0.0 - Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 5e-5:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = x / (0.0 - math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 5e-5)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(x / Float64(0.0 - tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 5e-5)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = x / (0.0 - tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 5e-5], 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[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.00000000000000024e-5

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   5. Simplified 52.8%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]
   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 52.8%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + 2 \cdot x\right)}} - x}{B}} \]

   if 5.00000000000000024e-5 < 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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 63.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 64.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 64.1%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.5e-47)
     (- (/ -1.0 B) t_0)
     (if (<= F 1.3e-27) (/ x (- 0.0 (tan B))) (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.5e-47) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.3e-27) {
		tmp = x / (0.0 - tan(B));
	} else {
		tmp = (1.0 / 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 <= (-6.5d-47)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.3d-27) then
        tmp = x / (0.0d0 - tan(b))
    else
        tmp = (1.0d0 / 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 <= -6.5e-47) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.3e-27) {
		tmp = x / (0.0 - Math.tan(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.5e-47:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.3e-27:
		tmp = x / (0.0 - math.tan(B))
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.5e-47)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.3e-27)
		tmp = Float64(x / Float64(0.0 - tan(B)));
	else
		tmp = Float64(Float64(1.0 / 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 <= -6.5e-47)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.3e-27)
		tmp = x / (0.0 - tan(B));
	else
		tmp = (1.0 / 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, -6.5e-47], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.3e-27], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.5000000000000004e-47

   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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 53.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 53.3%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(1\right)}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{B}\right)}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 69.3%

       \[\leadsto \color{blue}{\frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 68.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -6.5000000000000004e-47 < F < 1.30000000000000009e-27

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 68.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 68.4%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 68.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

   if 1.30000000000000009e-27 < F

   1. Initial program 56.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 93.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 74.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.8e-49) (- (/ -1.0 B) (/ x (tan B))) (/ x (- 0.0 (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-49) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = x / (0.0 - tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.8d-49)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = x / (0.0d0 - tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-49) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = x / (0.0 - Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.8e-49:
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = x / (0.0 - math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.8e-49)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(x / Float64(0.0 - tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.8e-49)
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = x / (0.0 - tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.8e-49], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.8e-49

   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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 53.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 53.3%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(1\right)}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{B}\right)}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 69.3%

       \[\leadsto \color{blue}{\frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 68.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -5.8e-49 < F

   1. Initial program 83.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 65.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 65.2%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 65.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 65.3%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 47.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.803 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{-1}{\frac{1}{F \cdot F} + \left(1 + \frac{x}{F \cdot F}\right)} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.803e-6)
   (/ (- (/ -1.0 (+ (/ 1.0 (* F F)) (+ 1.0 (/ x (* F F))))) x) B)
   (/ x (- 0.0 (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.803e-6) {
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B;
	} else {
		tmp = x / (0.0 - tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 1.803d-6) then
        tmp = (((-1.0d0) / ((1.0d0 / (f * f)) + (1.0d0 + (x / (f * f))))) - x) / b
    else
        tmp = x / (0.0d0 - tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.803e-6) {
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B;
	} else {
		tmp = x / (0.0 - Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1.803e-6:
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B
	else:
		tmp = x / (0.0 - math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.803e-6)
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(1.0 / Float64(F * F)) + Float64(1.0 + Float64(x / Float64(F * F))))) - x) / B);
	else
		tmp = Float64(x / Float64(0.0 - tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1.803e-6)
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B;
	else
		tmp = x / (0.0 - tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.803e-6], N[(N[(N[(-1.0 / N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(x / N[(0.0 - N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.8029999999999999e-6

   1. Initial program 75.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(\left(2 + x \cdot 2\right) + F \cdot F\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(x \cdot 2 + F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(2 \cdot x + F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(F \cdot F + 2 \cdot x\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(F \cdot F + 2 \cdot x\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. sqrt-pow1 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{{\left(2 + \left(F \cdot F + 2 \cdot x\right)\right)}^{-1}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}\right), \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   8. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + 2 \cdot x\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(-1 \cdot \left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)}\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\mathsf{neg}\left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) \cdot F\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) \cdot \left(-1 \cdot F\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right), \left(-1 \cdot F\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Simplified 71.6%

       \[\leadsto \frac{\frac{F}{\color{blue}{\left(1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) \cdot \left(0 - F\right)}}}{\sin B} - \frac{x}{\tan B} \]

   12. Taylor expanded in B around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}}{B}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}}{B}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}\right)\right)}{\color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}\right)\right)\right), \color{blue}{B}\right) \]

   14. Simplified 40.9%

       \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{1}{F \cdot F} + \left(1 + \frac{x}{F \cdot F}\right)} - x}{B}} \]

   if 1.8029999999999999e-6 < B

   1. Initial program 88.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   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. 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-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 62.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   9. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 63.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Applied egg-rr 63.0%

       \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.803 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{-1}{\frac{1}{F \cdot F} + \left(1 + \frac{x}{F \cdot F}\right)} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0 - \tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 37.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{-1}{\frac{1}{F \cdot F} + \left(1 + \frac{x}{F \cdot F}\right)} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 9.2e-10)
   (/ (- (/ -1.0 (+ (/ 1.0 (* F F)) (+ 1.0 (/ x (* F F))))) x) B)
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 9.2e-10) {
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B;
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 9.2d-10) then
        tmp = (((-1.0d0) / ((1.0d0 / (f * f)) + (1.0d0 + (x / (f * f))))) - x) / b
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 9.2e-10) {
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B;
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 9.2e-10:
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 9.2e-10)
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(1.0 / Float64(F * F)) + Float64(1.0 + Float64(x / Float64(F * F))))) - x) / B);
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 9.2e-10)
		tmp = ((-1.0 / ((1.0 / (F * F)) + (1.0 + (x / (F * F))))) - x) / B;
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 9.2e-10], N[(N[(N[(-1.0 / N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.20000000000000028e-10

   1. Initial program 76.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(\left(2 + x \cdot 2\right) + F \cdot F\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(x \cdot 2 + F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(2 \cdot x + F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(F \cdot F + 2 \cdot x\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(2 + \left(F \cdot F + 2 \cdot x\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. sqrt-pow1 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{{\left(2 + \left(F \cdot F + 2 \cdot x\right)\right)}^{-1}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} \cdot F}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}}\right), \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   8. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + 2 \cdot x\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(-1 \cdot \left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)}\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\mathsf{neg}\left(F \cdot \left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\mathsf{neg}\left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) \cdot F\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) \cdot \left(-1 \cdot F\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(\left(1 + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right), \left(-1 \cdot F\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   11. Simplified 71.2%

       \[\leadsto \frac{\frac{F}{\color{blue}{\left(1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) \cdot \left(0 - F\right)}}}{\sin B} - \frac{x}{\tan B} \]

   12. Taylor expanded in B around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}}{B}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}}{B}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}\right)\right)}{\color{blue}{B}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x + \frac{1}{1 + \left(\frac{1}{{F}^{2}} + \frac{x}{{F}^{2}}\right)}\right)\right)\right), \color{blue}{B}\right) \]

   14. Simplified 40.3%

       \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{1}{F \cdot F} + \left(1 + \frac{x}{F \cdot F}\right)} - x}{B}} \]

   if 9.20000000000000028e-10 < B

   1. Initial program 86.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 63.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 63.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 20.6%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 44.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{-1}{B}}{1 + -0.16666666666666666 \cdot \left(B \cdot B\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.8e-39)
   (- (/ (/ -1.0 B) (+ 1.0 (* -0.16666666666666666 (* B B)))) (/ x B))
   (if (<= F 4.2e-26)
     (* x (- (/ -1.0 B) (* B -0.3333333333333333)))
     (/
      (+
       (* (* B B) (+ 0.16666666666666666 (* x 0.3333333333333333)))
       (- 1.0 x))
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-39) {
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B);
	} else if (F <= 4.2e-26) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} 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 <= (-4.8d-39)) then
        tmp = (((-1.0d0) / b) / (1.0d0 + ((-0.16666666666666666d0) * (b * b)))) - (x / b)
    else if (f <= 4.2d-26) then
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    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 <= -4.8e-39) {
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B);
	} else if (F <= 4.2e-26) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} 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 <= -4.8e-39:
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B)
	elif F <= 4.2e-26:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	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 <= -4.8e-39)
		tmp = Float64(Float64(Float64(-1.0 / B) / Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B)))) - Float64(x / B));
	elseif (F <= 4.2e-26)
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	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 <= -4.8e-39)
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B);
	elseif (F <= 4.2e-26)
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	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, -4.8e-39], N[(N[(N[(-1.0 / B), $MachinePrecision] / N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-26], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $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 < -4.80000000000000031e-39

   1. Initial program 70.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 53.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 53.3%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(1\right)}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{B}\right)}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-lowering-*.f64 70.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 37.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   13. Simplified 37.9%

       \[\leadsto \frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666} - \color{blue}{\frac{x}{B}} \]

   if -4.80000000000000031e-39 < F < 4.20000000000000016e-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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 80.1%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \color{blue}{\left(\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\left(x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)\right), \color{blue}{B}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)\right)\right), B\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\frac{-1}{3} \cdot {B}^{2}\right) \cdot x\right)\right), B\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot {B}^{2}\right), x\right)\right), B\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left({B}^{2} \cdot \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 39.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

   10. Simplified 39.5%

       \[\leadsto \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \color{blue}{\frac{x + \left(\left(B \cdot B\right) \cdot -0.3333333333333333\right) \cdot x}{B}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{B} + \frac{-1}{3} \cdot B\right)\right)\right)\right) \]

       5. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3} \cdot B\right)\right)}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{B} + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3} \cdot B}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-1}{B} + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3}} \cdot B\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{B}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3} \cdot B\right)\right)}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3} \cdot B}\right)\right)\right)\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\left(\frac{-1}{3} \cdot B\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\left(B \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 26.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(B, \frac{-1}{3}\right)\right)\right)\right) \]

   13. Simplified 26.4%

       \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{B} + \left(-B \cdot -0.3333333333333333\right)\right)} \]

   if 4.20000000000000016e-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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\frac{-1}{2}}}{\sin B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-+r- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - x \cdot -2\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) - -2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \sin B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1}{F \cdot \sin B}\right)}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(F \cdot \sin B\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \left(\sin B \cdot F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 95.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), F\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   9. Simplified 95.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   11. 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. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

       4. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right), B\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + \left(1 - x\right)\right), B\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right), \left(1 - x\right)\right), B\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{3} \cdot x\right)\right)\right), \left(1 - x\right)\right), B\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \left(1 - x\right)\right), B\right) \]

       13. --lowering--.f64 48.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right), B\right) \]

   12. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{-1}{B}}{1 + -0.16666666666666666 \cdot \left(B \cdot B\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \left(1 - x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 44.5% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{-1}{B}}{1 + -0.16666666666666666 \cdot \left(B \cdot B\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.65 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.25e-37)
   (- (/ (/ -1.0 B) (+ 1.0 (* -0.16666666666666666 (* B B)))) (/ x B))
   (if (<= F 4.65e-45)
     (* x (- (/ -1.0 B) (* B -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.25e-37) {
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B);
	} else if (F <= 4.65e-45) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} 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 <= (-1.25d-37)) then
        tmp = (((-1.0d0) / b) / (1.0d0 + ((-0.16666666666666666d0) * (b * b)))) - (x / b)
    else if (f <= 4.65d-45) then
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    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 <= -1.25e-37) {
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B);
	} else if (F <= 4.65e-45) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.25e-37:
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B)
	elif F <= 4.65e-45:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.25e-37)
		tmp = Float64(Float64(Float64(-1.0 / B) / Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B)))) - Float64(x / B));
	elseif (F <= 4.65e-45)
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	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 <= -1.25e-37)
		tmp = ((-1.0 / B) / (1.0 + (-0.16666666666666666 * (B * B)))) - (x / B);
	elseif (F <= 4.65e-45)
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.25e-37], N[(N[(N[(-1.0 / B), $MachinePrecision] / N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.65e-45], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2499999999999999e-37

   1. Initial program 70.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 53.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 53.3%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(1\right)}{B}}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{B}\right)}{1 + \frac{-1}{6} \cdot {B}^{2}}\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{B}\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      12. *-lowering-*.f64 70.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 37.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   13. Simplified 37.9%

       \[\leadsto \frac{\frac{-1}{B}}{1 + \left(B \cdot B\right) \cdot -0.16666666666666666} - \color{blue}{\frac{x}{B}} \]

   if -1.2499999999999999e-37 < F < 4.65000000000000014e-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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 82.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 82.1%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \color{blue}{\left(\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\left(x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)\right), \color{blue}{B}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)\right)\right), B\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\frac{-1}{3} \cdot {B}^{2}\right) \cdot x\right)\right), B\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot {B}^{2}\right), x\right)\right), B\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left({B}^{2} \cdot \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 40.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

   10. Simplified 40.9%

       \[\leadsto \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \color{blue}{\frac{x + \left(\left(B \cdot B\right) \cdot -0.3333333333333333\right) \cdot x}{B}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{B} + \frac{-1}{3} \cdot B\right)\right)\right)\right) \]

       5. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3} \cdot B\right)\right)}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{B} + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3} \cdot B}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-1}{B} + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3}} \cdot B\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{B}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3} \cdot B\right)\right)}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3} \cdot B}\right)\right)\right)\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\left(\frac{-1}{3} \cdot B\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\left(B \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 27.3%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(B, \frac{-1}{3}\right)\right)\right)\right) \]

   13. Simplified 27.3%

       \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{B} + \left(-B \cdot -0.3333333333333333\right)\right)} \]

   if 4.65000000000000014e-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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 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 45.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   8. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{-1}{B}}{1 + -0.16666666666666666 \cdot \left(B \cdot B\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.65 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 44.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-40)
   (/ (- -1.0 x) B)
   (if (<= F 1.12e-45)
     (* x (- (/ -1.0 B) (* B -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-40) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.12e-45) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-40)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.12d-45) then
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-40) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.12e-45) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-40:
		tmp = (-1.0 - x) / B
	elif F <= 1.12e-45:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-40)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.12e-45)
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-40)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.12e-45)
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-40], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.12e-45], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e-40

   1. Initial program 70.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around -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 37.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -2.3e-40 < F < 1.1199999999999999e-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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \color{blue}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {B}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(\left(B \cdot B\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \left(B \cdot \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \left(B \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 82.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 82.1%

      \[\leadsto \frac{F}{\color{blue}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)}} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \color{blue}{\left(\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\left(x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)\right), \color{blue}{B}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)\right)\right), B\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\frac{-1}{3} \cdot {B}^{2}\right) \cdot x\right)\right), B\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot {B}^{2}\right), x\right)\right), B\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left({B}^{2} \cdot \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

      8. *-lowering-*.f64 40.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{*.f64}\left(B, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{-1}{3}\right), x\right)\right), B\right)\right) \]

   10. Simplified 40.9%

       \[\leadsto \frac{F}{B \cdot \left(1 + B \cdot \left(B \cdot -0.16666666666666666\right)\right)} \cdot {\left(F \cdot F + \left(2 - x \cdot -2\right)\right)}^{-0.5} - \color{blue}{\frac{x + \left(\left(B \cdot B\right) \cdot -0.3333333333333333\right) \cdot x}{B}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{B} + \frac{-1}{3} \cdot B\right)\right)\right)\right) \]

       5. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{B}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3} \cdot B\right)\right)}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{B} + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3} \cdot B}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-1}{B} + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3}} \cdot B\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{B}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{3} \cdot B\right)\right)}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{3} \cdot B}\right)\right)\right)\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\left(\frac{-1}{3} \cdot B\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\left(B \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 27.3%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, B\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(B, \frac{-1}{3}\right)\right)\right)\right) \]

   13. Simplified 27.3%

       \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{B} + \left(-B \cdot -0.3333333333333333\right)\right)} \]

   if 1.1199999999999999e-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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + 2 \cdot x\right)}} - x}{B}} \]

   6. Taylor expanded in F around 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 45.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   8. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 44.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.5e-55)
   (/ (- -1.0 x) B)
   (if (<= F 1.7e-23) (- 0.0 (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e-55) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.7e-23) {
		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 <= (-8.5d-55)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.7d-23) 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 <= -8.5e-55) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.7e-23) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.5e-55:
		tmp = (-1.0 - x) / B
	elif F <= 1.7e-23:
		tmp = 0.0 - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.5e-55)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.7e-23)
		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 <= -8.5e-55)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.7e-23)
		tmp = 0.0 - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.5e-55], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.7e-23], 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 < -8.49999999999999968e-55

   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

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   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 -8.49999999999999968e-55 < F < 1.7e-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

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   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 1.7e-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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 37.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-54) (/ (- -1.0 x) B) (- 0.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-54) {
		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 <= (-1.45d-54)) 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 <= -1.45e-54) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 0.0 - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-54:
		tmp = (-1.0 - x) / B
	else:
		tmp = 0.0 - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-54)
		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 <= -1.45e-54)
		tmp = (-1.0 - x) / B;
	else
		tmp = 0.0 - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-54], 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 < -1.45000000000000007e-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

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   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 -1.45000000000000007e-54 < 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

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 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

      \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

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 29: 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

      \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

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))))))