VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.2% → 99.7%

Time: 24.6s

Alternatives: 24

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: 76.2% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6400000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \left(\frac{1}{\sin B} + \frac{-1 - x}{\sin B \cdot {F}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 6400000.0)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) t_0)
       (+
        (* x (/ -1.0 (tan B)))
        (+ (/ 1.0 (sin B)) (/ (- -1.0 x) (* (sin B) (pow F 2.0)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 6400000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + ((1.0 / sin(B)) + ((-1.0 - x) / (sin(B) * pow(F, 2.0))));
	}
	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 <= (-8000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 6400000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + ((1.0d0 / sin(b)) + (((-1.0d0) - x) / (sin(b) * (f ** 2.0d0))))
    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 <= -8000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 6400000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + ((1.0 / Math.sin(B)) + ((-1.0 - x) / (Math.sin(B) * Math.pow(F, 2.0))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 6400000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + ((1.0 / math.sin(B)) + ((-1.0 - x) / (math.sin(B) * math.pow(F, 2.0))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 6400000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(1.0 / sin(B)) + Float64(Float64(-1.0 - x) / Float64(sin(B) * (F ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -8000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 6400000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + ((1.0 / sin(B)) + ((-1.0 - x) / (sin(B) * (F ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6400000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e9

   1. Initial program 63.4%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. expm1-log1p-u 66.2%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 66.2%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 66.2%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 66.2%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 99.9%

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

   6. Simplified 99.9%

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

   if -8e9 < F < 6.4e6

   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. div-inv 37.4%

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

      2. expm1-log1p-u 18.5%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 18.5%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   3. Applied egg-rr 54.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 18.5%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 37.4%

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

   5. Simplified 99.6%

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

   if 6.4e6 < F

   1. Initial program 58.4%

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

   2. Taylor expanded in F around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-*r/ 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. associate-*r* 99.8%

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

      6. metadata-eval 99.8%

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

      7. neg-mul-1 99.8%

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

   4. Simplified 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} + \frac{-1 + \left(-x\right)}{{F}^{2} \cdot \sin B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6400000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \left(\frac{1}{\sin B} + \frac{-1 - x}{\sin B \cdot {F}^{2}}\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (+
    (* x (/ -1.0 (tan B)))
    (/ F (* (sin B) (hypot F (sqrt (fma 2.0 x 2.0))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * hypot(F, sqrt(fma(2.0, x, 2.0)))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * hypot(F, sqrt(fma(2.0, x, 2.0))))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[F ^ 2 + N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 63.1%

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

   2. Taylor expanded in F around -inf 99.5%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 38.2%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 38.2%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 38.2%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 38.2%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

   if -1 < x

   1. Initial program 79.1%

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

      1. add-sqr-sqrt 79.1%

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

      2. unpow-prod-down 79.0%

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

      3. +-commutative 79.0%

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

      4. fma-def 79.0%

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

      5. fma-def 79.0%

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

      6. metadata-eval 79.0%

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

      7. metadata-eval 79.0%

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

      8. +-commutative 79.0%

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

      9. fma-def 79.0%

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

      10. fma-def 79.0%

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

      11. metadata-eval 79.0%

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

      12. metadata-eval 79.0%

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

   3. Applied egg-rr 79.0%

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

      1. pow-sqr 79.1%

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

      2. metadata-eval 79.1%

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

      3. unpow-1 79.1%

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

      4. fma-udef 79.1%

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

      5. +-commutative 79.1%

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

      6. fma-udef 79.1%

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

      7. unpow2 79.1%

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

      8. associate-+r+ 79.1%

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

      9. unpow2 79.1%

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

      10. unpow1 79.1%

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

      11. sqr-pow 79.1%

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

      12. hypot-def 89.0%

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

      13. +-commutative 89.0%

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

      14. fma-def 89.0%

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

      15. metadata-eval 89.0%

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

   5. Simplified 89.0%

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

      1. *-commutative 89.0%

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

      2. frac-times 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. unpow1/2 99.6%

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

   7. Applied egg-rr 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right) \cdot \sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7800000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+37}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7800000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7e+37)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7800000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7e+37) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-7800000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 7d+37) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    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 <= -7800000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 7e+37) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -7800000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 7e+37:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7800000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7e+37)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -7800000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 7e+37)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7800000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7e+37], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.8e9

   1. Initial program 63.4%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. expm1-log1p-u 66.2%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 66.2%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 66.2%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 66.2%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 99.9%

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

   6. Simplified 99.9%

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

   if -7.8e9 < F < 7e37

   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. div-inv 38.1%

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

      2. expm1-log1p-u 17.9%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 17.9%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   3. Applied egg-rr 53.6%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 17.9%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 38.1%

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

   5. Simplified 99.6%

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

   if 7e37 < F

   1. Initial program 56.2%

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

   2. Taylor expanded in F around inf 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7800000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+37}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.00033:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-14}:\\ \;\;\;\;t_0 + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.00033)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 9.4e-14)
       (+ t_0 (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.00033) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 9.4e-14) {
		tmp = t_0 + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.00033d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 9.4d-14) then
        tmp = t_0 + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.00033) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 9.4e-14) {
		tmp = t_0 + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.00033:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 9.4e-14:
		tmp = t_0 + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.00033)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 9.4e-14)
		tmp = Float64(t_0 + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.00033)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 9.4e-14)
		tmp = t_0 + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00033], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.4e-14], N[(t$95$0 + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3e-4

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 99.0%

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

      1. div-inv 99.1%

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

      2. expm1-log1p-u 64.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 64.0%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 64.0%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 64.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 99.1%

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

   6. Simplified 99.1%

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

   if -3.3e-4 < F < 9.4000000000000003e-14

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. unpow-prod-down 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. fma-def 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. +-commutative 99.4%

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

      9. fma-def 99.4%

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

      10. fma-def 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. pow-sqr 99.4%

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

      2. metadata-eval 99.4%

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

      3. unpow-1 99.4%

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

      4. fma-udef 99.4%

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

      5. +-commutative 99.4%

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

      6. fma-udef 99.4%

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

      7. unpow2 99.4%

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

      8. associate-+r+ 99.4%

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

      9. unpow2 99.4%

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

      10. unpow1 99.4%

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

      11. sqr-pow 99.4%

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

      12. hypot-def 99.4%

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

      13. +-commutative 99.4%

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

      14. fma-def 99.4%

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

      15. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. unpow1/2 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in F around 0 99.5%

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

   if 9.4000000000000003e-14 < F

   1. Initial program 60.4%

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

   2. Taylor expanded in F around inf 98.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00033:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 45:\\ \;\;\;\;t_0 + t_1 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
   (if (<= F -1.18e-20)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -8.2e-134)
       (- (* (/ F (sin B)) t_1) (/ x B))
       (if (<= F 45.0) (+ t_0 (* t_1 (/ F B))) (+ t_0 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -1.18e-20) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -8.2e-134) {
		tmp = ((F / sin(B)) * t_1) - (x / B);
	} else if (F <= 45.0) {
		tmp = t_0 + (t_1 * (F / B));
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = ((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)
    if (f <= (-1.18d-20)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-8.2d-134)) then
        tmp = ((f / sin(b)) * t_1) - (x / b)
    else if (f <= 45.0d0) then
        tmp = t_0 + (t_1 * (f / b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -1.18e-20) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -8.2e-134) {
		tmp = ((F / Math.sin(B)) * t_1) - (x / B);
	} else if (F <= 45.0) {
		tmp = t_0 + (t_1 * (F / B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)
	tmp = 0
	if F <= -1.18e-20:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -8.2e-134:
		tmp = ((F / math.sin(B)) * t_1) - (x / B)
	elif F <= 45.0:
		tmp = t_0 + (t_1 * (F / B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5
	tmp = 0.0
	if (F <= -1.18e-20)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -8.2e-134)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_1) - Float64(x / B));
	elseif (F <= 45.0)
		tmp = Float64(t_0 + Float64(t_1 * Float64(F / B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = ((2.0 + (F * F)) + (x * 2.0)) ^ -0.5;
	tmp = 0.0;
	if (F <= -1.18e-20)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -8.2e-134)
		tmp = ((F / sin(B)) * t_1) - (x / B);
	elseif (F <= 45.0)
		tmp = t_0 + (t_1 * (F / B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -1.18e-20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.2e-134], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 45.0], N[(t$95$0 + N[(t$95$1 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.1800000000000001e-20

   1. Initial program 67.2%

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

   2. Taylor expanded in F around -inf 97.9%

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

      1. div-inv 98.0%

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

      2. expm1-log1p-u 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 63.7%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 63.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 98.0%

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

   6. Simplified 98.0%

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

   if -1.1800000000000001e-20 < F < -8.2000000000000004e-134

   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. Taylor expanded in B around 0 92.0%

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

   if -8.2000000000000004e-134 < F < 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. Taylor expanded in B around 0 90.6%

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

   if 45 < F

   1. Initial program 59.4%

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

   2. Taylor expanded in F around inf 98.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 45:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-135}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-14}:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.18e-20)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -3.1e-135)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 9.4e-14)
         (+ t_0 (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
         (+ t_0 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.18e-20) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -3.1e-135) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 9.4e-14) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.18d-20)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-3.1d-135)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 9.4d-14) then
        tmp = t_0 + ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.18e-20) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -3.1e-135) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 9.4e-14) {
		tmp = t_0 + ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.18e-20:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -3.1e-135:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 9.4e-14:
		tmp = t_0 + ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.18e-20)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -3.1e-135)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 9.4e-14)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.18e-20)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -3.1e-135)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 9.4e-14)
		tmp = t_0 + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.18e-20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.1e-135], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.4e-14], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.1800000000000001e-20

   1. Initial program 67.2%

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

   2. Taylor expanded in F around -inf 97.9%

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

      1. div-inv 98.0%

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

      2. expm1-log1p-u 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 63.7%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 63.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 98.0%

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

   6. Simplified 98.0%

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

   if -1.1800000000000001e-20 < F < -3.1000000000000001e-135

   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. Taylor expanded in B around 0 92.0%

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

   if -3.1000000000000001e-135 < F < 9.4000000000000003e-14

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 90.4%

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

   3. Taylor expanded in F around 0 90.4%

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

   if 9.4000000000000003e-14 < F

   1. Initial program 60.4%

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

   2. Taylor expanded in F around inf 98.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-135}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.00033:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-14}:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.00033)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 9.4e-14)
       (+ t_0 (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.00033) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 9.4e-14) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.00033d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 9.4d-14) then
        tmp = t_0 + ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.00033) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 9.4e-14) {
		tmp = t_0 + ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.00033:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 9.4e-14:
		tmp = t_0 + ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.00033)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 9.4e-14)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.00033)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 9.4e-14)
		tmp = t_0 + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00033], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.4e-14], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3e-4

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 99.0%

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

      1. div-inv 99.1%

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

      2. expm1-log1p-u 64.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 64.0%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 64.0%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 64.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 99.1%

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

   6. Simplified 99.1%

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

   if -3.3e-4 < F < 9.4000000000000003e-14

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 84.5%

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

   3. Taylor expanded in F around 0 84.5%

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

   if 9.4000000000000003e-14 < F

   1. Initial program 60.4%

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

   2. Taylor expanded in F around inf 98.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00033:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 8: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(x \cdot 2 + {F}^{2}\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.8e-21)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.55e-221)
     (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* x 2.0) (pow F 2.0)))))) x) B)
     (if (<= F 3.9e-84)
       (/ (- x) (/ (sin B) (cos B)))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-21) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.55e-221) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((x * 2.0) + pow(F, 2.0)))))) - x) / B;
	} else if (F <= 3.9e-84) {
		tmp = -x / (sin(B) / cos(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.8d-21)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.55d-221)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((x * 2.0d0) + (f ** 2.0d0)))))) - x) / b
    else if (f <= 3.9d-84) then
        tmp = -x / (sin(b) / cos(b))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-21) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.55e-221) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((x * 2.0) + Math.pow(F, 2.0)))))) - x) / B;
	} else if (F <= 3.9e-84) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.8e-21:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.55e-221:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((x * 2.0) + math.pow(F, 2.0)))))) - x) / B
	elif F <= 3.9e-84:
		tmp = -x / (math.sin(B) / math.cos(B))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.8e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.55e-221)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(x * 2.0) + (F ^ 2.0)))))) - x) / B);
	elseif (F <= 3.9e-84)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.8e-21)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.55e-221)
		tmp = ((F * sqrt((1.0 / (2.0 + ((x * 2.0) + (F ^ 2.0)))))) - x) / B;
	elseif (F <= 3.9e-84)
		tmp = -x / (sin(B) / cos(B));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.8e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.55e-221], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(x * 2.0), $MachinePrecision] + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.9e-84], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.7999999999999999e-21

   1. Initial program 67.2%

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

   2. Taylor expanded in F around -inf 97.9%

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

      1. div-inv 98.0%

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

      2. expm1-log1p-u 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 63.7%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 63.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 98.0%

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

   6. Simplified 98.0%

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

   if -4.7999999999999999e-21 < F < -1.55e-221

   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. Taylor expanded in B around 0 73.3%

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

   3. Taylor expanded in B around 0 61.7%

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

   4. Taylor expanded in B around 0 61.7%

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

   if -1.55e-221 < F < 3.90000000000000023e-84

   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. Taylor expanded in F around -inf 33.7%

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

   3. Taylor expanded in x around inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-/l* 77.0%

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

      3. distribute-neg-frac 77.0%

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

   5. Simplified 77.0%

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

   if 3.90000000000000023e-84 < F

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf 92.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(x \cdot 2 + {F}^{2}\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9: 71.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.95 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 63:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+149} \lor \neg \left(F \leq 1.75 \cdot 10^{+225}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -1.95e+177)
     t_0
     (if (<= F -1.5e+31)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -9.5e-21)
         t_0
         (if (<= F -1.55e-221)
           (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
           (if (<= F 63.0)
             (/ (- x) (/ (sin B) (cos B)))
             (if (or (<= F 1.15e+149) (not (<= F 1.75e+225)))
               (- (/ F (* (sin B) F)) (/ x B))
               (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -1.95e+177) {
		tmp = t_0;
	} else if (F <= -1.5e+31) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9.5e-21) {
		tmp = t_0;
	} else if (F <= -1.55e-221) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 63.0) {
		tmp = -x / (sin(B) / cos(B));
	} else if ((F <= 1.15e+149) || !(F <= 1.75e+225)) {
		tmp = (F / (sin(B) * F)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-1.95d+177)) then
        tmp = t_0
    else if (f <= (-1.5d+31)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9.5d-21)) then
        tmp = t_0
    else if (f <= (-1.55d-221)) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 63.0d0) then
        tmp = -x / (sin(b) / cos(b))
    else if ((f <= 1.15d+149) .or. (.not. (f <= 1.75d+225))) then
        tmp = (f / (sin(b) * f)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -1.95e+177) {
		tmp = t_0;
	} else if (F <= -1.5e+31) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9.5e-21) {
		tmp = t_0;
	} else if (F <= -1.55e-221) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 63.0) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if ((F <= 1.15e+149) || !(F <= 1.75e+225)) {
		tmp = (F / (Math.sin(B) * F)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -1.95e+177:
		tmp = t_0
	elif F <= -1.5e+31:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9.5e-21:
		tmp = t_0
	elif F <= -1.55e-221:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 63.0:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif (F <= 1.15e+149) or not (F <= 1.75e+225):
		tmp = (F / (math.sin(B) * F)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -1.95e+177)
		tmp = t_0;
	elseif (F <= -1.5e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9.5e-21)
		tmp = t_0;
	elseif (F <= -1.55e-221)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 63.0)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif ((F <= 1.15e+149) || !(F <= 1.75e+225))
		tmp = Float64(Float64(F / Float64(sin(B) * F)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -1.95e+177)
		tmp = t_0;
	elseif (F <= -1.5e+31)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9.5e-21)
		tmp = t_0;
	elseif (F <= -1.55e-221)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 63.0)
		tmp = -x / (sin(B) / cos(B));
	elseif ((F <= 1.15e+149) || ~((F <= 1.75e+225)))
		tmp = (F / (sin(B) * F)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.95e+177], t$95$0, If[LessEqual[F, -1.5e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.5e-21], t$95$0, If[LessEqual[F, -1.55e-221], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 63.0], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.15e+149], N[Not[LessEqual[F, 1.75e+225]], $MachinePrecision]], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -1.95e177 or -1.49999999999999995e31 < F < -9.4999999999999994e-21

   1. Initial program 64.2%

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

   2. Taylor expanded in F around -inf 96.5%

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

   3. Taylor expanded in B around 0 87.0%

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

   if -1.95e177 < F < -1.49999999999999995e31

   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. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 84.2%

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

   4. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-neg-frac 84.2%

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

      5. metadata-eval 84.2%

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

      6. sub-neg 84.2%

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

   6. Simplified 84.2%

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

   if -9.4999999999999994e-21 < F < -1.55e-221

   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. Taylor expanded in B around 0 73.3%

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

   3. Taylor expanded in B around 0 61.7%

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

   4. Taylor expanded in F around 0 61.7%

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

   if -1.55e-221 < F < 63

   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. Taylor expanded in F around -inf 37.2%

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

   3. Taylor expanded in x around inf 71.7%

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

      1. mul-1-neg 71.7%

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

      2. associate-/l* 71.8%

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

      3. distribute-neg-frac 71.8%

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

   5. Simplified 71.8%

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

   if 63 < F < 1.1499999999999999e149 or 1.7500000000000002e225 < F

   1. Initial program 60.8%

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

      1. add-sqr-sqrt 60.8%

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

      2. unpow-prod-down 60.6%

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

      3. +-commutative 60.6%

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

      4. fma-def 60.6%

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

      5. fma-def 60.6%

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

      6. metadata-eval 60.6%

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

      7. metadata-eval 60.6%

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

      8. +-commutative 60.6%

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

      9. fma-def 60.6%

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

      10. fma-def 60.6%

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

      11. metadata-eval 60.6%

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

      12. metadata-eval 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. pow-sqr 60.8%

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

      2. metadata-eval 60.8%

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

      3. unpow-1 60.8%

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

      4. fma-udef 60.8%

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

      5. +-commutative 60.8%

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

      6. fma-udef 60.8%

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

      7. unpow2 60.8%

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

      8. associate-+r+ 60.8%

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

      9. unpow2 60.8%

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

      10. unpow1 60.8%

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

      11. sqr-pow 57.0%

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

      12. hypot-def 71.8%

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

      13. +-commutative 71.8%

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

      14. fma-def 71.8%

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

      15. metadata-eval 71.8%

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

   5. Simplified 71.8%

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

      1. *-commutative 71.8%

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

      2. frac-times 89.0%

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

      3. *-un-lft-identity 89.0%

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

      4. unpow1/2 87.2%

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

   7. Applied egg-rr 87.2%

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

   8. Taylor expanded in F around inf 98.4%

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

      1. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in B around 0 86.6%

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

   if 1.1499999999999999e149 < F < 1.7500000000000002e225

   1. Initial program 54.5%

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

      1. add-sqr-sqrt 54.5%

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

      2. unpow-prod-down 54.5%

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

      3. +-commutative 54.5%

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

      4. fma-def 54.5%

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

      5. fma-def 54.5%

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

      6. metadata-eval 54.5%

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

      7. metadata-eval 54.5%

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

      8. +-commutative 54.5%

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

      9. fma-def 54.5%

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

      10. fma-def 54.5%

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

      11. metadata-eval 54.5%

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

      12. metadata-eval 54.5%

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

   3. Applied egg-rr 54.5%

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

      1. pow-sqr 54.5%

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

      2. metadata-eval 54.5%

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

      3. unpow-1 54.5%

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

      4. fma-udef 54.5%

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

      5. +-commutative 54.5%

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

      6. fma-udef 54.5%

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

      7. unpow2 54.5%

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

      8. associate-+r+ 54.5%

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

      9. unpow2 54.5%

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

      10. unpow1 54.5%

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

      11. sqr-pow 29.6%

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

      12. hypot-def 43.1%

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

      13. +-commutative 43.1%

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

      14. fma-def 43.1%

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

      15. metadata-eval 43.1%

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

   5. Simplified 43.1%

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

      1. *-commutative 43.1%

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

      2. frac-times 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. unpow1/2 74.7%

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

   7. Applied egg-rr 74.7%

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

   8. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in B around 0 91.9%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.95 \cdot 10^{+177}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 63:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+149} \lor \neg \left(F \leq 1.75 \cdot 10^{+225}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 10: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-21)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.55e-221)
     (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
     (if (<= F 2.5e-84)
       (/ (- x) (/ (sin B) (cos B)))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-21) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.55e-221) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 2.5e-84) {
		tmp = -x / (sin(B) / cos(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.15d-21)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.55d-221)) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 2.5d-84) then
        tmp = -x / (sin(b) / cos(b))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-21) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.55e-221) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 2.5e-84) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-21:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.55e-221:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 2.5e-84:
		tmp = -x / (math.sin(B) / math.cos(B))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.55e-221)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 2.5e-84)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.15e-21)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.55e-221)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 2.5e-84)
		tmp = -x / (sin(B) / cos(B));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.55e-221], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-84], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.15e-21

   1. Initial program 67.2%

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

   2. Taylor expanded in F around -inf 97.9%

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

      1. div-inv 98.0%

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

      2. expm1-log1p-u 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 63.7%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 63.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 98.0%

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

   6. Simplified 98.0%

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

   if -1.15e-21 < F < -1.55e-221

   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. Taylor expanded in B around 0 73.3%

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

   3. Taylor expanded in B around 0 61.7%

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

   4. Taylor expanded in F around 0 61.7%

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

   if -1.55e-221 < F < 2.5000000000000001e-84

   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. Taylor expanded in F around -inf 33.7%

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

   3. Taylor expanded in x around inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-/l* 77.0%

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

      3. distribute-neg-frac 77.0%

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

   5. Simplified 77.0%

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

   if 2.5000000000000001e-84 < F

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf 92.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 11: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 63:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+145} \lor \neg \left(F \leq 5.2 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e-22)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.55e-221)
     (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
     (if (<= F 63.0)
       (/ (- x) (/ (sin B) (cos B)))
       (if (or (<= F 3.5e+145) (not (<= F 5.2e+224)))
         (- (/ F (* (sin B) F)) (/ x B))
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-22) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.55e-221) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 63.0) {
		tmp = -x / (sin(B) / cos(B));
	} else if ((F <= 3.5e+145) || !(F <= 5.2e+224)) {
		tmp = (F / (sin(B) * F)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.1d-22)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.55d-221)) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 63.0d0) then
        tmp = -x / (sin(b) / cos(b))
    else if ((f <= 3.5d+145) .or. (.not. (f <= 5.2d+224))) then
        tmp = (f / (sin(b) * f)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-22) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.55e-221) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 63.0) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if ((F <= 3.5e+145) || !(F <= 5.2e+224)) {
		tmp = (F / (Math.sin(B) * F)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e-22:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.55e-221:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 63.0:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif (F <= 3.5e+145) or not (F <= 5.2e+224):
		tmp = (F / (math.sin(B) * F)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e-22)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.55e-221)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 63.0)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif ((F <= 3.5e+145) || !(F <= 5.2e+224))
		tmp = Float64(Float64(F / Float64(sin(B) * F)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e-22)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.55e-221)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 63.0)
		tmp = -x / (sin(B) / cos(B));
	elseif ((F <= 3.5e+145) || ~((F <= 5.2e+224)))
		tmp = (F / (sin(B) * F)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e-22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.55e-221], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 63.0], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 3.5e+145], N[Not[LessEqual[F, 5.2e+224]], $MachinePrecision]], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.10000000000000013e-22

   1. Initial program 67.2%

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

   2. Taylor expanded in F around -inf 97.9%

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

      1. div-inv 98.0%

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

      2. expm1-log1p-u 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      3. expm1-udef 63.7%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]

   4. Applied egg-rr 63.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{-1}{\sin B} \]
   5. Step-by-step derivation

      1. expm1-def 63.7%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{-1}{\sin B} \]

      2. expm1-log1p 98.0%

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

   6. Simplified 98.0%

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

   if -3.10000000000000013e-22 < F < -1.55e-221

   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. Taylor expanded in B around 0 73.3%

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

   3. Taylor expanded in B around 0 61.7%

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

   4. Taylor expanded in F around 0 61.7%

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

   if -1.55e-221 < F < 63

   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. Taylor expanded in F around -inf 37.2%

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

   3. Taylor expanded in x around inf 71.7%

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

      1. mul-1-neg 71.7%

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

      2. associate-/l* 71.8%

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

      3. distribute-neg-frac 71.8%

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

   5. Simplified 71.8%

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

   if 63 < F < 3.5000000000000001e145 or 5.2000000000000001e224 < F

   1. Initial program 60.8%

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

      1. add-sqr-sqrt 60.8%

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

      2. unpow-prod-down 60.6%

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

      3. +-commutative 60.6%

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

      4. fma-def 60.6%

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

      5. fma-def 60.6%

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

      6. metadata-eval 60.6%

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

      7. metadata-eval 60.6%

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

      8. +-commutative 60.6%

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

      9. fma-def 60.6%

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

      10. fma-def 60.6%

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

      11. metadata-eval 60.6%

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

      12. metadata-eval 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. pow-sqr 60.8%

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

      2. metadata-eval 60.8%

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

      3. unpow-1 60.8%

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

      4. fma-udef 60.8%

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

      5. +-commutative 60.8%

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

      6. fma-udef 60.8%

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

      7. unpow2 60.8%

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

      8. associate-+r+ 60.8%

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

      9. unpow2 60.8%

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

      10. unpow1 60.8%

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

      11. sqr-pow 57.0%

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

      12. hypot-def 71.8%

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

      13. +-commutative 71.8%

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

      14. fma-def 71.8%

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

      15. metadata-eval 71.8%

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

   5. Simplified 71.8%

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

      1. *-commutative 71.8%

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

      2. frac-times 89.0%

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

      3. *-un-lft-identity 89.0%

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

      4. unpow1/2 87.2%

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

   7. Applied egg-rr 87.2%

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

   8. Taylor expanded in F around inf 98.4%

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

      1. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in B around 0 86.6%

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

   if 3.5000000000000001e145 < F < 5.2000000000000001e224

   1. Initial program 54.5%

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

      1. add-sqr-sqrt 54.5%

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

      2. unpow-prod-down 54.5%

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

      3. +-commutative 54.5%

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

      4. fma-def 54.5%

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

      5. fma-def 54.5%

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

      6. metadata-eval 54.5%

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

      7. metadata-eval 54.5%

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

      8. +-commutative 54.5%

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

      9. fma-def 54.5%

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

      10. fma-def 54.5%

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

      11. metadata-eval 54.5%

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

      12. metadata-eval 54.5%

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

   3. Applied egg-rr 54.5%

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

      1. pow-sqr 54.5%

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

      2. metadata-eval 54.5%

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

      3. unpow-1 54.5%

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

      4. fma-udef 54.5%

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

      5. +-commutative 54.5%

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

      6. fma-udef 54.5%

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

      7. unpow2 54.5%

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

      8. associate-+r+ 54.5%

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

      9. unpow2 54.5%

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

      10. unpow1 54.5%

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

      11. sqr-pow 29.6%

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

      12. hypot-def 43.1%

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

      13. +-commutative 43.1%

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

      14. fma-def 43.1%

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

      15. metadata-eval 43.1%

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

   5. Simplified 43.1%

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

      1. *-commutative 43.1%

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

      2. frac-times 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. unpow1/2 74.7%

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

   7. Applied egg-rr 74.7%

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

   8. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in B around 0 91.9%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 63:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+145} \lor \neg \left(F \leq 5.2 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 12: 64.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ t_2 := \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.32 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-221}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-180}:\\ \;\;\;\;t_1 + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+224}:\\ \;\;\;\;t_1 + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (* x (/ 1.0 (tan B)))))
        (t_1 (* x (/ -1.0 (tan B))))
        (t_2 (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))))
   (if (<= F -1.3e+177)
     t_0
     (if (<= F -1.32e+31)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -4.8e-21)
         t_0
         (if (<= F -1.3e-221)
           t_2
           (if (<= F 2.7e-180)
             (+ t_1 (* (/ F B) (/ -1.0 F)))
             (if (<= F 1.15e-59)
               t_2
               (if (<= F 7.5e+224)
                 (+ t_1 (/ 1.0 B))
                 (- (/ F (* (sin B) F)) (/ x B)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double t_1 = x * (-1.0 / tan(B));
	double t_2 = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double tmp;
	if (F <= -1.3e+177) {
		tmp = t_0;
	} else if (F <= -1.32e+31) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.8e-21) {
		tmp = t_0;
	} else if (F <= -1.3e-221) {
		tmp = t_2;
	} else if (F <= 2.7e-180) {
		tmp = t_1 + ((F / B) * (-1.0 / F));
	} else if (F <= 1.15e-59) {
		tmp = t_2;
	} else if (F <= 7.5e+224) {
		tmp = t_1 + (1.0 / B);
	} else {
		tmp = (F / (sin(B) * F)) - (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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    t_1 = x * ((-1.0d0) / tan(b))
    t_2 = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    if (f <= (-1.3d+177)) then
        tmp = t_0
    else if (f <= (-1.32d+31)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.8d-21)) then
        tmp = t_0
    else if (f <= (-1.3d-221)) then
        tmp = t_2
    else if (f <= 2.7d-180) then
        tmp = t_1 + ((f / b) * ((-1.0d0) / f))
    else if (f <= 1.15d-59) then
        tmp = t_2
    else if (f <= 7.5d+224) then
        tmp = t_1 + (1.0d0 / b)
    else
        tmp = (f / (sin(b) * f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double t_1 = x * (-1.0 / Math.tan(B));
	double t_2 = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double tmp;
	if (F <= -1.3e+177) {
		tmp = t_0;
	} else if (F <= -1.32e+31) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.8e-21) {
		tmp = t_0;
	} else if (F <= -1.3e-221) {
		tmp = t_2;
	} else if (F <= 2.7e-180) {
		tmp = t_1 + ((F / B) * (-1.0 / F));
	} else if (F <= 1.15e-59) {
		tmp = t_2;
	} else if (F <= 7.5e+224) {
		tmp = t_1 + (1.0 / B);
	} else {
		tmp = (F / (Math.sin(B) * F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	t_1 = x * (-1.0 / math.tan(B))
	t_2 = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	tmp = 0
	if F <= -1.3e+177:
		tmp = t_0
	elif F <= -1.32e+31:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.8e-21:
		tmp = t_0
	elif F <= -1.3e-221:
		tmp = t_2
	elif F <= 2.7e-180:
		tmp = t_1 + ((F / B) * (-1.0 / F))
	elif F <= 1.15e-59:
		tmp = t_2
	elif F <= 7.5e+224:
		tmp = t_1 + (1.0 / B)
	else:
		tmp = (F / (math.sin(B) * F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	t_2 = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B))
	tmp = 0.0
	if (F <= -1.3e+177)
		tmp = t_0;
	elseif (F <= -1.32e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.8e-21)
		tmp = t_0;
	elseif (F <= -1.3e-221)
		tmp = t_2;
	elseif (F <= 2.7e-180)
		tmp = Float64(t_1 + Float64(Float64(F / B) * Float64(-1.0 / F)));
	elseif (F <= 1.15e-59)
		tmp = t_2;
	elseif (F <= 7.5e+224)
		tmp = Float64(t_1 + Float64(1.0 / B));
	else
		tmp = Float64(Float64(F / Float64(sin(B) * F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	t_1 = x * (-1.0 / tan(B));
	t_2 = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	tmp = 0.0;
	if (F <= -1.3e+177)
		tmp = t_0;
	elseif (F <= -1.32e+31)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.8e-21)
		tmp = t_0;
	elseif (F <= -1.3e-221)
		tmp = t_2;
	elseif (F <= 2.7e-180)
		tmp = t_1 + ((F / B) * (-1.0 / F));
	elseif (F <= 1.15e-59)
		tmp = t_2;
	elseif (F <= 7.5e+224)
		tmp = t_1 + (1.0 / B);
	else
		tmp = (F / (sin(B) * F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.3e+177], t$95$0, If[LessEqual[F, -1.32e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.8e-21], t$95$0, If[LessEqual[F, -1.3e-221], t$95$2, If[LessEqual[F, 2.7e-180], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-59], t$95$2, If[LessEqual[F, 7.5e+224], N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -1.2999999999999999e177 or -1.32000000000000011e31 < F < -4.7999999999999999e-21

   1. Initial program 64.2%

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

   2. Taylor expanded in F around -inf 96.5%

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

   3. Taylor expanded in B around 0 87.0%

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

   if -1.2999999999999999e177 < F < -1.32000000000000011e31

   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. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 84.2%

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

   4. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-neg-frac 84.2%

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

      5. metadata-eval 84.2%

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

      6. sub-neg 84.2%

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

   6. Simplified 84.2%

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

   if -4.7999999999999999e-21 < F < -1.3000000000000001e-221 or 2.70000000000000014e-180 < F < 1.1499999999999999e-59

   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. Taylor expanded in B around 0 76.8%

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

   3. Taylor expanded in B around 0 62.1%

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

   4. Taylor expanded in F around 0 62.1%

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

   if -1.3000000000000001e-221 < F < 2.70000000000000014e-180

   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. Taylor expanded in B around 0 95.4%

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

   3. Taylor expanded in F around -inf 64.2%

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

   if 1.1499999999999999e-59 < F < 7.500000000000001e224

   1. Initial program 79.4%

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

      1. add-sqr-sqrt 79.3%

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

      2. unpow-prod-down 79.2%

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

      3. +-commutative 79.2%

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

      4. fma-def 79.2%

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

      5. fma-def 79.2%

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

      6. metadata-eval 79.2%

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

      7. metadata-eval 79.2%

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

      8. +-commutative 79.2%

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

      9. fma-def 79.2%

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

      10. fma-def 79.2%

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

      11. metadata-eval 79.2%

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

      12. metadata-eval 79.2%

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

   3. Applied egg-rr 79.2%

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

      1. pow-sqr 79.3%

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

      2. metadata-eval 79.3%

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

      3. unpow-1 79.3%

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

      4. fma-udef 79.3%

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

      5. +-commutative 79.3%

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

      6. fma-udef 79.3%

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

      7. unpow2 79.3%

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

      8. associate-+r+ 79.3%

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

      9. unpow2 79.3%

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

      10. unpow1 79.3%

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

      11. sqr-pow 67.3%

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

      12. hypot-def 72.1%

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

      13. +-commutative 72.1%

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

      14. fma-def 72.1%

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

      15. metadata-eval 72.1%

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

   5. Simplified 72.1%

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

      1. *-commutative 72.1%

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

      2. frac-times 86.3%

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

      3. *-un-lft-identity 86.3%

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

      4. unpow1/2 86.3%

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

   7. Applied egg-rr 86.3%

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

   8. Taylor expanded in F around inf 93.3%

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

      1. *-commutative 93.3%

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

   10. Simplified 93.3%

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

   11. Taylor expanded in B around 0 77.6%

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

   if 7.500000000000001e224 < F

   1. Initial program 13.3%

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

      1. add-sqr-sqrt 13.3%

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

      2. unpow-prod-down 13.3%

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

      3. +-commutative 13.3%

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

      4. fma-def 13.3%

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

      5. fma-def 13.3%

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

      6. metadata-eval 13.3%

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

      7. metadata-eval 13.3%

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

      8. +-commutative 13.3%

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

      9. fma-def 13.3%

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

      10. fma-def 13.3%

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

      11. metadata-eval 13.3%

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

      12. metadata-eval 13.3%

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

   3. Applied egg-rr 13.3%

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

      1. pow-sqr 13.3%

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

      2. metadata-eval 13.3%

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

      3. unpow-1 13.3%

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

      4. fma-udef 13.3%

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

      5. +-commutative 13.3%

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

      6. fma-udef 13.3%

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

      7. unpow2 13.3%

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

      8. associate-+r+ 13.3%

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

      9. unpow2 13.3%

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

      10. unpow1 13.3%

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

      11. sqr-pow 13.3%

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

      12. hypot-def 51.1%

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

      13. +-commutative 51.1%

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

      14. fma-def 51.1%

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

      15. metadata-eval 51.1%

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

   5. Simplified 51.1%

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

      1. *-commutative 51.1%

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

      2. frac-times 86.2%

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

      3. *-un-lft-identity 86.2%

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

      4. unpow1/2 81.6%

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

   7. Applied egg-rr 81.6%

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

   8. Taylor expanded in F around inf 99.8%

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

      1. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 90.8%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B \cdot F} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+177}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.32 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-221}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-59}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+224}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 61.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -2.15 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.6 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+38} \lor \neg \left(F \leq 1.8 \cdot 10^{+146}\right) \land F \leq 3.8 \cdot 10^{+224}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -2.15e+177)
     t_0
     (if (<= F -8.8e+30)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -6.6e-49)
         t_0
         (if (<= F -5.2e-159)
           (/ (- x) B)
           (if (or (<= F 2e+38) (and (not (<= F 1.8e+146)) (<= F 3.8e+224)))
             (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))
             (- (/ F (* (sin B) F)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -2.15e+177) {
		tmp = t_0;
	} else if (F <= -8.8e+30) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -6.6e-49) {
		tmp = t_0;
	} else if (F <= -5.2e-159) {
		tmp = -x / B;
	} else if ((F <= 2e+38) || (!(F <= 1.8e+146) && (F <= 3.8e+224))) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	} else {
		tmp = (F / (sin(B) * F)) - (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) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-2.15d+177)) then
        tmp = t_0
    else if (f <= (-8.8d+30)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-6.6d-49)) then
        tmp = t_0
    else if (f <= (-5.2d-159)) then
        tmp = -x / b
    else if ((f <= 2d+38) .or. (.not. (f <= 1.8d+146)) .and. (f <= 3.8d+224)) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    else
        tmp = (f / (sin(b) * f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -2.15e+177) {
		tmp = t_0;
	} else if (F <= -8.8e+30) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -6.6e-49) {
		tmp = t_0;
	} else if (F <= -5.2e-159) {
		tmp = -x / B;
	} else if ((F <= 2e+38) || (!(F <= 1.8e+146) && (F <= 3.8e+224))) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	} else {
		tmp = (F / (Math.sin(B) * F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -2.15e+177:
		tmp = t_0
	elif F <= -8.8e+30:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -6.6e-49:
		tmp = t_0
	elif F <= -5.2e-159:
		tmp = -x / B
	elif (F <= 2e+38) or (not (F <= 1.8e+146) and (F <= 3.8e+224)):
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	else:
		tmp = (F / (math.sin(B) * F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -2.15e+177)
		tmp = t_0;
	elseif (F <= -8.8e+30)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -6.6e-49)
		tmp = t_0;
	elseif (F <= -5.2e-159)
		tmp = Float64(Float64(-x) / B);
	elseif ((F <= 2e+38) || (!(F <= 1.8e+146) && (F <= 3.8e+224)))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	else
		tmp = Float64(Float64(F / Float64(sin(B) * F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -2.15e+177)
		tmp = t_0;
	elseif (F <= -8.8e+30)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -6.6e-49)
		tmp = t_0;
	elseif (F <= -5.2e-159)
		tmp = -x / B;
	elseif ((F <= 2e+38) || (~((F <= 1.8e+146)) && (F <= 3.8e+224)))
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	else
		tmp = (F / (sin(B) * F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.15e+177], t$95$0, If[LessEqual[F, -8.8e+30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.6e-49], t$95$0, If[LessEqual[F, -5.2e-159], N[((-x) / B), $MachinePrecision], If[Or[LessEqual[F, 2e+38], And[N[Not[LessEqual[F, 1.8e+146]], $MachinePrecision], LessEqual[F, 3.8e+224]]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.15000000000000018e177 or -8.7999999999999999e30 < F < -6.6e-49

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 94.7%

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

   3. Taylor expanded in B around 0 85.9%

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

   if -2.15000000000000018e177 < F < -8.7999999999999999e30

   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. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 84.2%

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

   4. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-neg-frac 84.2%

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

      5. metadata-eval 84.2%

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

      6. sub-neg 84.2%

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

   6. Simplified 84.2%

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

   if -6.6e-49 < F < -5.1999999999999997e-159

   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. Taylor expanded in F around -inf 20.2%

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

   3. Taylor expanded in B around 0 12.2%

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

      1. associate-*r/ 12.2%

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

      2. distribute-lft-in 12.2%

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

      3. metadata-eval 12.2%

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

      4. mul-1-neg 12.2%

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

   5. Simplified 12.2%

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

   6. Taylor expanded in x around inf 36.9%

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

      1. associate-*r/ 36.9%

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

      2. mul-1-neg 36.9%

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

   8. Simplified 36.9%

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

   if -5.1999999999999997e-159 < F < 1.99999999999999995e38 or 1.7999999999999999e146 < F < 3.80000000000000025e224

   1. Initial program 88.8%

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

      1. add-sqr-sqrt 88.8%

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

      2. unpow-prod-down 88.8%

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

      3. +-commutative 88.8%

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

      4. fma-def 88.8%

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

      5. fma-def 88.8%

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

      6. metadata-eval 88.8%

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

      7. metadata-eval 88.8%

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

      8. +-commutative 88.8%

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

      9. fma-def 88.8%

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

      10. fma-def 88.8%

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

      11. metadata-eval 88.8%

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

      12. metadata-eval 88.8%

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

   3. Applied egg-rr 88.8%

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

      1. pow-sqr 88.8%

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

      2. metadata-eval 88.8%

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

      3. unpow-1 88.8%

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

      4. fma-udef 88.8%

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

      5. +-commutative 88.8%

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

      6. fma-udef 88.8%

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

      7. unpow2 88.8%

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

      8. associate-+r+ 88.8%

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

      9. unpow2 88.8%

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

      10. unpow1 88.8%

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

      11. sqr-pow 81.9%

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

      12. hypot-def 85.1%

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

      13. +-commutative 85.1%

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

      14. fma-def 85.1%

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

      15. metadata-eval 85.1%

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

   5. Simplified 85.1%

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

      1. *-commutative 85.1%

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

      2. frac-times 92.7%

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

      3. *-un-lft-identity 92.7%

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

      4. unpow1/2 92.7%

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

   7. Applied egg-rr 92.7%

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

   8. Taylor expanded in F around inf 48.6%

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

      1. *-commutative 48.6%

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

   10. Simplified 48.6%

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

   11. Taylor expanded in B around 0 60.1%

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

   if 1.99999999999999995e38 < F < 1.7999999999999999e146 or 3.80000000000000025e224 < F

   1. Initial program 57.0%

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

      1. add-sqr-sqrt 57.0%

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

      2. unpow-prod-down 56.8%

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

      3. +-commutative 56.8%

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

      4. fma-def 56.8%

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

      5. fma-def 56.8%

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

      6. metadata-eval 56.8%

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

      7. metadata-eval 56.8%

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

      8. +-commutative 56.8%

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

      9. fma-def 56.8%

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

      10. fma-def 56.8%

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

      11. metadata-eval 56.8%

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

      12. metadata-eval 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. pow-sqr 57.0%

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

      2. metadata-eval 57.0%

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

      3. unpow-1 57.0%

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

      4. fma-udef 57.0%

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

      5. +-commutative 57.0%

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

      6. fma-udef 57.0%

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

      7. unpow2 57.0%

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

      8. associate-+r+ 57.0%

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

      9. unpow2 57.0%

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

      10. unpow1 57.0%

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

      11. sqr-pow 54.8%

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

      12. hypot-def 71.1%

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

      13. +-commutative 71.1%

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

      14. fma-def 71.1%

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

      15. metadata-eval 71.1%

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

   5. Simplified 71.1%

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

      1. *-commutative 71.1%

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

      2. frac-times 89.9%

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

      3. *-un-lft-identity 89.9%

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

      4. unpow1/2 87.9%

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

   7. Applied egg-rr 87.9%

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

   8. Taylor expanded in F around inf 99.7%

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

      1. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in B around 0 88.5%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B \cdot F} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{+177}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+38} \lor \neg \left(F \leq 1.8 \cdot 10^{+146}\right) \land F \leq 3.8 \cdot 10^{+224}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \end{array} \]

Alternative 14: 63.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 46:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+147} \lor \neg \left(F \leq 8.5 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (- (/ -1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -3.5e+176)
     t_1
     (if (<= F -1.6e+31)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -4.5e-49)
         t_1
         (if (<= F -3e-159)
           (/ (- x) B)
           (if (<= F 46.0)
             (+ t_0 (* (/ F B) (/ -1.0 F)))
             (if (or (<= F 7.8e+147) (not (<= F 8.5e+224)))
               (- (/ F (* (sin B) F)) (/ x B))
               (+ t_0 (/ 1.0 B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -3.5e+176) {
		tmp = t_1;
	} else if (F <= -1.6e+31) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.5e-49) {
		tmp = t_1;
	} else if (F <= -3e-159) {
		tmp = -x / B;
	} else if (F <= 46.0) {
		tmp = t_0 + ((F / B) * (-1.0 / F));
	} else if ((F <= 7.8e+147) || !(F <= 8.5e+224)) {
		tmp = (F / (sin(B) * F)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-3.5d+176)) then
        tmp = t_1
    else if (f <= (-1.6d+31)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.5d-49)) then
        tmp = t_1
    else if (f <= (-3d-159)) then
        tmp = -x / b
    else if (f <= 46.0d0) then
        tmp = t_0 + ((f / b) * ((-1.0d0) / f))
    else if ((f <= 7.8d+147) .or. (.not. (f <= 8.5d+224))) then
        tmp = (f / (sin(b) * f)) - (x / b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -3.5e+176) {
		tmp = t_1;
	} else if (F <= -1.6e+31) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.5e-49) {
		tmp = t_1;
	} else if (F <= -3e-159) {
		tmp = -x / B;
	} else if (F <= 46.0) {
		tmp = t_0 + ((F / B) * (-1.0 / F));
	} else if ((F <= 7.8e+147) || !(F <= 8.5e+224)) {
		tmp = (F / (Math.sin(B) * F)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -3.5e+176:
		tmp = t_1
	elif F <= -1.6e+31:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.5e-49:
		tmp = t_1
	elif F <= -3e-159:
		tmp = -x / B
	elif F <= 46.0:
		tmp = t_0 + ((F / B) * (-1.0 / F))
	elif (F <= 7.8e+147) or not (F <= 8.5e+224):
		tmp = (F / (math.sin(B) * F)) - (x / B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -3.5e+176)
		tmp = t_1;
	elseif (F <= -1.6e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.5e-49)
		tmp = t_1;
	elseif (F <= -3e-159)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 46.0)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(-1.0 / F)));
	elseif ((F <= 7.8e+147) || !(F <= 8.5e+224))
		tmp = Float64(Float64(F / Float64(sin(B) * F)) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = (-1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -3.5e+176)
		tmp = t_1;
	elseif (F <= -1.6e+31)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.5e-49)
		tmp = t_1;
	elseif (F <= -3e-159)
		tmp = -x / B;
	elseif (F <= 46.0)
		tmp = t_0 + ((F / B) * (-1.0 / F));
	elseif ((F <= 7.8e+147) || ~((F <= 8.5e+224)))
		tmp = (F / (sin(B) * F)) - (x / B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e+176], t$95$1, If[LessEqual[F, -1.6e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.5e-49], t$95$1, If[LessEqual[F, -3e-159], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 46.0], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 7.8e+147], N[Not[LessEqual[F, 8.5e+224]], $MachinePrecision]], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -3.50000000000000003e176 or -1.6e31 < F < -4.5000000000000002e-49

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 94.7%

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

   3. Taylor expanded in B around 0 85.9%

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

   if -3.50000000000000003e176 < F < -1.6e31

   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. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 84.2%

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

   4. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-neg-frac 84.2%

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

      5. metadata-eval 84.2%

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

      6. sub-neg 84.2%

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

   6. Simplified 84.2%

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

   if -4.5000000000000002e-49 < F < -3.00000000000000009e-159

   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. Taylor expanded in F around -inf 20.2%

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

   3. Taylor expanded in B around 0 12.2%

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

      1. associate-*r/ 12.2%

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

      2. distribute-lft-in 12.2%

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

      3. metadata-eval 12.2%

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

      4. mul-1-neg 12.2%

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

   5. Simplified 12.2%

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

   6. Taylor expanded in x around inf 36.9%

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

      1. associate-*r/ 36.9%

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

      2. mul-1-neg 36.9%

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

   8. Simplified 36.9%

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

   if -3.00000000000000009e-159 < F < 46

   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. Taylor expanded in B around 0 91.0%

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

   3. Taylor expanded in F around -inf 53.2%

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

   if 46 < F < 7.80000000000000033e147 or 8.50000000000000046e224 < F

   1. Initial program 60.8%

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

      1. add-sqr-sqrt 60.8%

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

      2. unpow-prod-down 60.6%

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

      3. +-commutative 60.6%

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

      4. fma-def 60.6%

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

      5. fma-def 60.6%

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

      6. metadata-eval 60.6%

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

      7. metadata-eval 60.6%

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

      8. +-commutative 60.6%

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

      9. fma-def 60.6%

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

      10. fma-def 60.6%

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

      11. metadata-eval 60.6%

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

      12. metadata-eval 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. pow-sqr 60.8%

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

      2. metadata-eval 60.8%

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

      3. unpow-1 60.8%

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

      4. fma-udef 60.8%

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

      5. +-commutative 60.8%

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

      6. fma-udef 60.8%

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

      7. unpow2 60.8%

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

      8. associate-+r+ 60.8%

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

      9. unpow2 60.8%

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

      10. unpow1 60.8%

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

      11. sqr-pow 57.0%

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

      12. hypot-def 71.8%

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

      13. +-commutative 71.8%

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

      14. fma-def 71.8%

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

      15. metadata-eval 71.8%

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

   5. Simplified 71.8%

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

      1. *-commutative 71.8%

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

      2. frac-times 89.0%

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

      3. *-un-lft-identity 89.0%

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

      4. unpow1/2 87.2%

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

   7. Applied egg-rr 87.2%

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

   8. Taylor expanded in F around inf 98.4%

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

      1. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in B around 0 86.6%

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

   if 7.80000000000000033e147 < F < 8.50000000000000046e224

   1. Initial program 54.5%

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

      1. add-sqr-sqrt 54.5%

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

      2. unpow-prod-down 54.5%

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

      3. +-commutative 54.5%

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

      4. fma-def 54.5%

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

      5. fma-def 54.5%

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

      6. metadata-eval 54.5%

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

      7. metadata-eval 54.5%

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

      8. +-commutative 54.5%

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

      9. fma-def 54.5%

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

      10. fma-def 54.5%

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

      11. metadata-eval 54.5%

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

      12. metadata-eval 54.5%

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

   3. Applied egg-rr 54.5%

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

      1. pow-sqr 54.5%

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

      2. metadata-eval 54.5%

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

      3. unpow-1 54.5%

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

      4. fma-udef 54.5%

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

      5. +-commutative 54.5%

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

      6. fma-udef 54.5%

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

      7. unpow2 54.5%

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

      8. associate-+r+ 54.5%

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

      9. unpow2 54.5%

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

      10. unpow1 54.5%

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

      11. sqr-pow 29.6%

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

      12. hypot-def 43.1%

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

      13. +-commutative 43.1%

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

      14. fma-def 43.1%

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

      15. metadata-eval 43.1%

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

   5. Simplified 43.1%

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

      1. *-commutative 43.1%

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

      2. frac-times 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. unpow1/2 74.7%

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

   7. Applied egg-rr 74.7%

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

   8. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in B around 0 91.9%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{+176}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 46:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+147} \lor \neg \left(F \leq 8.5 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 15: 62.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+148} \lor \neg \left(F \leq 5.7 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (- (/ -1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -3e+176)
     t_1
     (if (<= F -9e+30)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -1.7e-48)
         t_1
         (if (<= F -4.6e-159)
           (/ (- x) B)
           (if (<= F 2.5e+38)
             (+ t_0 (* (/ F B) (/ 1.0 F)))
             (if (or (<= F 9.5e+148) (not (<= F 5.7e+224)))
               (- (/ F (* (sin B) F)) (/ x B))
               (+ t_0 (/ 1.0 B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -3e+176) {
		tmp = t_1;
	} else if (F <= -9e+30) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.7e-48) {
		tmp = t_1;
	} else if (F <= -4.6e-159) {
		tmp = -x / B;
	} else if (F <= 2.5e+38) {
		tmp = t_0 + ((F / B) * (1.0 / F));
	} else if ((F <= 9.5e+148) || !(F <= 5.7e+224)) {
		tmp = (F / (sin(B) * F)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-3d+176)) then
        tmp = t_1
    else if (f <= (-9d+30)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.7d-48)) then
        tmp = t_1
    else if (f <= (-4.6d-159)) then
        tmp = -x / b
    else if (f <= 2.5d+38) then
        tmp = t_0 + ((f / b) * (1.0d0 / f))
    else if ((f <= 9.5d+148) .or. (.not. (f <= 5.7d+224))) then
        tmp = (f / (sin(b) * f)) - (x / b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -3e+176) {
		tmp = t_1;
	} else if (F <= -9e+30) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.7e-48) {
		tmp = t_1;
	} else if (F <= -4.6e-159) {
		tmp = -x / B;
	} else if (F <= 2.5e+38) {
		tmp = t_0 + ((F / B) * (1.0 / F));
	} else if ((F <= 9.5e+148) || !(F <= 5.7e+224)) {
		tmp = (F / (Math.sin(B) * F)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -3e+176:
		tmp = t_1
	elif F <= -9e+30:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.7e-48:
		tmp = t_1
	elif F <= -4.6e-159:
		tmp = -x / B
	elif F <= 2.5e+38:
		tmp = t_0 + ((F / B) * (1.0 / F))
	elif (F <= 9.5e+148) or not (F <= 5.7e+224):
		tmp = (F / (math.sin(B) * F)) - (x / B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -3e+176)
		tmp = t_1;
	elseif (F <= -9e+30)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.7e-48)
		tmp = t_1;
	elseif (F <= -4.6e-159)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 2.5e+38)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / F)));
	elseif ((F <= 9.5e+148) || !(F <= 5.7e+224))
		tmp = Float64(Float64(F / Float64(sin(B) * F)) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = (-1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -3e+176)
		tmp = t_1;
	elseif (F <= -9e+30)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.7e-48)
		tmp = t_1;
	elseif (F <= -4.6e-159)
		tmp = -x / B;
	elseif (F <= 2.5e+38)
		tmp = t_0 + ((F / B) * (1.0 / F));
	elseif ((F <= 9.5e+148) || ~((F <= 5.7e+224)))
		tmp = (F / (sin(B) * F)) - (x / B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3e+176], t$95$1, If[LessEqual[F, -9e+30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.7e-48], t$95$1, If[LessEqual[F, -4.6e-159], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 2.5e+38], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 9.5e+148], N[Not[LessEqual[F, 5.7e+224]], $MachinePrecision]], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -3e176 or -8.9999999999999999e30 < F < -1.70000000000000014e-48

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 94.7%

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

   3. Taylor expanded in B around 0 85.9%

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

   if -3e176 < F < -8.9999999999999999e30

   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. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 84.2%

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

   4. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-neg-frac 84.2%

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

      5. metadata-eval 84.2%

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

      6. sub-neg 84.2%

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

   6. Simplified 84.2%

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

   if -1.70000000000000014e-48 < F < -4.59999999999999957e-159

   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. Taylor expanded in F around -inf 20.2%

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

   3. Taylor expanded in B around 0 12.2%

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

      1. associate-*r/ 12.2%

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

      2. distribute-lft-in 12.2%

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

      3. metadata-eval 12.2%

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

      4. mul-1-neg 12.2%

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

   5. Simplified 12.2%

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

   6. Taylor expanded in x around inf 36.9%

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

      1. associate-*r/ 36.9%

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

      2. mul-1-neg 36.9%

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

   8. Simplified 36.9%

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

   if -4.59999999999999957e-159 < F < 2.49999999999999985e38

   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. Taylor expanded in B around 0 90.3%

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

   3. Taylor expanded in F around inf 55.4%

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

   if 2.49999999999999985e38 < F < 9.5000000000000002e148 or 5.69999999999999997e224 < F

   1. Initial program 57.0%

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

      1. add-sqr-sqrt 57.0%

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

      2. unpow-prod-down 56.8%

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

      3. +-commutative 56.8%

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

      4. fma-def 56.8%

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

      5. fma-def 56.8%

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

      6. metadata-eval 56.8%

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

      7. metadata-eval 56.8%

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

      8. +-commutative 56.8%

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

      9. fma-def 56.8%

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

      10. fma-def 56.8%

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

      11. metadata-eval 56.8%

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

      12. metadata-eval 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. pow-sqr 57.0%

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

      2. metadata-eval 57.0%

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

      3. unpow-1 57.0%

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

      4. fma-udef 57.0%

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

      5. +-commutative 57.0%

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

      6. fma-udef 57.0%

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

      7. unpow2 57.0%

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

      8. associate-+r+ 57.0%

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

      9. unpow2 57.0%

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

      10. unpow1 57.0%

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

      11. sqr-pow 54.8%

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

      12. hypot-def 71.1%

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

      13. +-commutative 71.1%

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

      14. fma-def 71.1%

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

      15. metadata-eval 71.1%

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

   5. Simplified 71.1%

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

      1. *-commutative 71.1%

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

      2. frac-times 89.9%

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

      3. *-un-lft-identity 89.9%

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

      4. unpow1/2 87.9%

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

   7. Applied egg-rr 87.9%

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

   8. Taylor expanded in F around inf 99.7%

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

      1. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in B around 0 88.5%

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

   if 9.5000000000000002e148 < F < 5.69999999999999997e224

   1. Initial program 54.5%

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

      1. add-sqr-sqrt 54.5%

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

      2. unpow-prod-down 54.5%

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

      3. +-commutative 54.5%

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

      4. fma-def 54.5%

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

      5. fma-def 54.5%

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

      6. metadata-eval 54.5%

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

      7. metadata-eval 54.5%

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

      8. +-commutative 54.5%

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

      9. fma-def 54.5%

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

      10. fma-def 54.5%

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

      11. metadata-eval 54.5%

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

      12. metadata-eval 54.5%

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

   3. Applied egg-rr 54.5%

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

      1. pow-sqr 54.5%

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

      2. metadata-eval 54.5%

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

      3. unpow-1 54.5%

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

      4. fma-udef 54.5%

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

      5. +-commutative 54.5%

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

      6. fma-udef 54.5%

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

      7. unpow2 54.5%

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

      8. associate-+r+ 54.5%

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

      9. unpow2 54.5%

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

      10. unpow1 54.5%

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

      11. sqr-pow 29.6%

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

      12. hypot-def 43.1%

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

      13. +-commutative 43.1%

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

      14. fma-def 43.1%

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

      15. metadata-eval 43.1%

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

   5. Simplified 43.1%

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

      1. *-commutative 43.1%

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

      2. frac-times 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. unpow1/2 74.7%

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

   7. Applied egg-rr 74.7%

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

   8. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in B around 0 91.9%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+176}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+148} \lor \neg \left(F \leq 5.7 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{F}{\sin B \cdot F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 16: 61.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -2.2e+177)
     t_0
     (if (<= F -9.2e+30)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -4.5e-49)
         t_0
         (if (<= F -4.4e-159)
           (/ (- x) B)
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -2.2e+177) {
		tmp = t_0;
	} else if (F <= -9.2e+30) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.5e-49) {
		tmp = t_0;
	} else if (F <= -4.4e-159) {
		tmp = -x / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-2.2d+177)) then
        tmp = t_0
    else if (f <= (-9.2d+30)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.5d-49)) then
        tmp = t_0
    else if (f <= (-4.4d-159)) then
        tmp = -x / b
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -2.2e+177) {
		tmp = t_0;
	} else if (F <= -9.2e+30) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.5e-49) {
		tmp = t_0;
	} else if (F <= -4.4e-159) {
		tmp = -x / B;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -2.2e+177:
		tmp = t_0
	elif F <= -9.2e+30:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.5e-49:
		tmp = t_0
	elif F <= -4.4e-159:
		tmp = -x / B
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -2.2e+177)
		tmp = t_0;
	elseif (F <= -9.2e+30)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.5e-49)
		tmp = t_0;
	elseif (F <= -4.4e-159)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -2.2e+177)
		tmp = t_0;
	elseif (F <= -9.2e+30)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.5e-49)
		tmp = t_0;
	elseif (F <= -4.4e-159)
		tmp = -x / B;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e+177], t$95$0, If[LessEqual[F, -9.2e+30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.5e-49], t$95$0, If[LessEqual[F, -4.4e-159], N[((-x) / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1999999999999998e177 or -9.2e30 < F < -4.5000000000000002e-49

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 94.7%

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

   3. Taylor expanded in B around 0 85.9%

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

   if -2.1999999999999998e177 < F < -9.2e30

   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. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 84.2%

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

   4. Taylor expanded in x around 0 84.2%

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

      1. sub-neg 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-neg-frac 84.2%

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

      5. metadata-eval 84.2%

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

      6. sub-neg 84.2%

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

   6. Simplified 84.2%

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

   if -4.5000000000000002e-49 < F < -4.4e-159

   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. Taylor expanded in F around -inf 20.2%

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

   3. Taylor expanded in B around 0 12.2%

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

      1. associate-*r/ 12.2%

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

      2. distribute-lft-in 12.2%

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

      3. metadata-eval 12.2%

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

      4. mul-1-neg 12.2%

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

   5. Simplified 12.2%

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

   6. Taylor expanded in x around inf 36.9%

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

      1. associate-*r/ 36.9%

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

      2. mul-1-neg 36.9%

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

   8. Simplified 36.9%

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

   if -4.4e-159 < F

   1. Initial program 78.1%

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

      1. add-sqr-sqrt 78.1%

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

      2. unpow-prod-down 78.1%

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

      3. +-commutative 78.1%

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

      4. fma-def 78.1%

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

      5. fma-def 78.1%

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

      6. metadata-eval 78.1%

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

      7. metadata-eval 78.1%

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

      8. +-commutative 78.1%

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

      9. fma-def 78.1%

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

      10. fma-def 78.1%

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

      11. metadata-eval 78.1%

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

      12. metadata-eval 78.1%

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

   3. Applied egg-rr 78.1%

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

      1. pow-sqr 78.1%

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

      2. metadata-eval 78.1%

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

      3. unpow-1 78.1%

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

      4. fma-udef 78.1%

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

      5. +-commutative 78.1%

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

      6. fma-udef 78.1%

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

      7. unpow2 78.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\left(\color{blue}{{F}^{2}} + 2\right) + 2 \cdot x}} \]

      8. associate-+r+ 78.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \]

      9. unpow2 78.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \]

      10. unpow1 78.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{1}}}} \]

      11. sqr-pow 72.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}} \]

      12. hypot-def 80.4%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(F, {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}\right)}} \]

      13. +-commutative 80.4%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(2 \cdot x + 2\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      14. fma-def 80.4%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(\mathsf{fma}\left(2, x, 2\right)\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      15. metadata-eval 80.4%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{\color{blue}{0.5}}\right)} \]

   5. Simplified 80.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 80.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)} \cdot \frac{F}{\sin B}} \]

      2. frac-times 91.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 \cdot F}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right) \cdot \sin B}} \]

      3. *-un-lft-identity 91.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F}}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right) \cdot \sin B} \]

      4. unpow1/2 91.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\mathsf{hypot}\left(F, \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}}\right) \cdot \sin B} \]

   7. Applied egg-rr 91.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right) \cdot \sin B}} \]

   8. Taylor expanded in F around inf 65.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{F \cdot \sin B}} \]
   9. Step-by-step derivation

      1. *-commutative 65.7%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{\sin B \cdot F}} \]

   10. Simplified 65.7%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{\sin B \cdot F}} \]

   11. Taylor expanded in B around 0 62.5%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{+177}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-159}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 17: 52.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= x -4.3e-57)
     t_0
     (if (<= x 4e-285)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= x 1.5e+19) (- (/ 1.0 B) (/ x B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (x <= -4.3e-57) {
		tmp = t_0;
	} else if (x <= 4e-285) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (x <= 1.5e+19) {
		tmp = (1.0 / B) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    if (x <= (-4.3d-57)) then
        tmp = t_0
    else if (x <= 4d-285) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (x <= 1.5d+19) then
        tmp = (1.0d0 / b) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (x <= -4.3e-57) {
		tmp = t_0;
	} else if (x <= 4e-285) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (x <= 1.5e+19) {
		tmp = (1.0 / B) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if x <= -4.3e-57:
		tmp = t_0
	elif x <= 4e-285:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif x <= 1.5e+19:
		tmp = (1.0 / B) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (x <= -4.3e-57)
		tmp = t_0;
	elseif (x <= 4e-285)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (x <= 1.5e+19)
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (x <= -4.3e-57)
		tmp = t_0;
	elseif (x <= 4e-285)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (x <= 1.5e+19)
		tmp = (1.0 / B) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e-57], t$95$0, If[LessEqual[x, 4e-285], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e+19], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.30000000000000022e-57 or 1.5e19 < x

   1. Initial program 82.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 91.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 94.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -4.30000000000000022e-57 < x < 4.0000000000000003e-285

   1. Initial program 70.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 39.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 39.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   4. Taylor expanded in x around 0 39.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. sub-neg 39.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{\sin B}\right)} \]

      2. mul-1-neg 39.1%

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right)} + \left(-\frac{1}{\sin B}\right) \]

      3. +-commutative 39.1%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{x}{B}\right)} \]

      4. distribute-neg-frac 39.1%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{x}{B}\right) \]

      5. metadata-eval 39.1%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{x}{B}\right) \]

      6. sub-neg 39.1%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   6. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if 4.0000000000000003e-285 < x < 1.5e19

   1. Initial program 75.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 50.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 39.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 26.5%

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 18: 52.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-51)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.25e-50) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-51) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.25e-50) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.2d-51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.25d-50) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-51) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.25e-50) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-51:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.25e-50:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.25e-50)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-51)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.25e-50)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.25e-50], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.19999999999999947e-51

   1. Initial program 69.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 94.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 64.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   4. Taylor expanded in x around 0 64.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. sub-neg 64.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{\sin B}\right)} \]

      2. mul-1-neg 64.7%

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right)} + \left(-\frac{1}{\sin B}\right) \]

      3. +-commutative 64.7%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{x}{B}\right)} \]

      4. distribute-neg-frac 64.7%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{x}{B}\right) \]

      5. metadata-eval 64.7%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{x}{B}\right) \]

      6. sub-neg 64.7%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   6. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -8.19999999999999947e-51 < F < 1.24999999999999992e-50

   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. Taylor expanded in F around -inf 31.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 16.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 16.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 16.4%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 16.4%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 16.4%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 35.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 35.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 35.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.24999999999999992e-50 < F

   1. Initial program 61.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 49.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 26.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 51.5%

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 19: 45.1% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.06 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.06e-44)
   (/ (- -1.0 x) B)
   (if (<= F 3.9e-49) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.06e-44) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.9e-49) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.06d-44)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.9d-49) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.06e-44) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.9e-49) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.06e-44:
		tmp = (-1.0 - x) / B
	elif F <= 3.9e-49:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.06e-44)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.9e-49)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.06e-44)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.9e-49)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.06e-44], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.9e-49], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0599999999999999e-44

   1. Initial program 68.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 96.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 43.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 43.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 43.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 43.9%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 43.9%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -1.0599999999999999e-44 < F < 3.90000000000000011e-49

   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. Taylor expanded in F around -inf 30.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 16.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 16.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 16.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 16.1%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 16.1%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 16.1%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 34.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 34.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 34.7%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 3.90000000000000011e-49 < F

   1. Initial program 61.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 49.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 26.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 51.5%

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.06 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 20: 45.1% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-47}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.05e-47)
   (- (* B -0.16666666666666666) (/ (+ x 1.0) B))
   (if (<= F 3.9e-49) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.05e-47) {
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B);
	} else if (F <= 3.9e-49) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.05d-47)) then
        tmp = (b * (-0.16666666666666666d0)) - ((x + 1.0d0) / b)
    else if (f <= 3.9d-49) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.05e-47) {
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B);
	} else if (F <= 3.9e-49) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.05e-47:
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B)
	elif F <= 3.9e-49:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.05e-47)
		tmp = Float64(Float64(B * -0.16666666666666666) - Float64(Float64(x + 1.0) / B));
	elseif (F <= 3.9e-49)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.05e-47)
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B);
	elseif (F <= 3.9e-49)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.05e-47], N[(N[(B * -0.16666666666666666), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e-49], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.05000000000000001e-47

   1. Initial program 68.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 96.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 66.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   4. Taylor expanded in B around 0 44.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + -0.16666666666666666 \cdot B} \]
   5. Step-by-step derivation

      1. +-commutative 44.1%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot B + -1 \cdot \frac{1 + x}{B}} \]

      2. mul-1-neg 44.1%

         \[\leadsto -0.16666666666666666 \cdot B + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

      3. unsub-neg 44.1%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot B - \frac{1 + x}{B}} \]

      4. *-commutative 44.1%

         \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{1 + x}{B} \]

   6. Simplified 44.1%

      \[\leadsto \color{blue}{B \cdot -0.16666666666666666 - \frac{1 + x}{B}} \]

   if -2.05000000000000001e-47 < F < 3.90000000000000011e-49

   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. Taylor expanded in F around -inf 30.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 16.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 16.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 16.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 16.1%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 16.1%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 16.1%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 34.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 34.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 34.7%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 3.90000000000000011e-49 < F

   1. Initial program 61.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 49.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 26.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 51.5%

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-47}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 21: 30.0% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{+109}:\\ \;\;\;\;\frac{x + -1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.65e+109) (/ (+ x -1.0) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.65e+109) {
		tmp = (x + -1.0) / B;
	} else {
		tmp = -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 <= (-2.65d+109)) then
        tmp = (x + (-1.0d0)) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.65e+109) {
		tmp = (x + -1.0) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.65e+109:
		tmp = (x + -1.0) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.65e+109)
		tmp = Float64(Float64(x + -1.0) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.65e+109)
		tmp = (x + -1.0) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.65e+109], N[(N[(x + -1.0), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.65000000000000013e109

   1. Initial program 52.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 50.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 50.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 50.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 50.8%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 50.8%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + \left(-x\right)}{B}\right)\right)} \]

      2. expm1-udef 23.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1 + \left(-x\right)}{B}\right)} - 1} \]

      3. +-commutative 23.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-x\right) + -1}}{B}\right)} - 1 \]

      4. add-sqr-sqrt 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + -1}{B}\right)} - 1 \]

      5. sqrt-unprod 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + -1}{B}\right)} - 1 \]

      6. sqr-neg 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}} + -1}{B}\right)} - 1 \]

      7. sqrt-unprod 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1}{B}\right)} - 1 \]

      8. add-sqr-sqrt 13.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x} + -1}{B}\right)} - 1 \]

   7. Applied egg-rr 13.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + -1}{B}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 13.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + -1}{B}\right)\right)} \]

      2. expm1-log1p 33.1%

         \[\leadsto \color{blue}{\frac{x + -1}{B}} \]

   9. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{x + -1}{B}} \]

   if -2.65000000000000013e109 < F

   1. Initial program 83.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 47.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 22.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 22.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 22.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 22.7%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 22.7%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 29.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 29.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 29.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{+109}:\\ \;\;\;\;\frac{x + -1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 22: 37.6% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.6e-45) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.6e-45) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -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 <= (-6.6d-45)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.6e-45) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.6e-45:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.6e-45)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.6e-45)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.6e-45], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.6000000000000001e-45

   1. Initial program 68.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 96.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 43.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 43.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 43.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 43.9%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 43.9%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -6.6000000000000001e-45 < F

   1. Initial program 81.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 39.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 20.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 20.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 20.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 20.9%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 20.9%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 20.9%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 30.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 30.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 30.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 23: 30.2% accurate, 53.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e+102) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+102) {
		tmp = -1.0 / B;
	} else {
		tmp = -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 <= (-7.2d+102)) then
        tmp = (-1.0d0) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+102) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e+102:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e+102)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e+102)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e+102], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -7.2000000000000003e102

   1. Initial program 52.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 50.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 50.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 50.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 50.8%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 50.8%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around 0 33.0%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -7.2000000000000003e102 < F

   1. Initial program 83.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 47.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 22.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 22.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 22.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 22.7%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. mul-1-neg 22.7%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 22.7%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 29.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 29.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 29.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 24: 10.6% 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 77.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 57.3%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in B around 0 28.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
4. Step-by-step derivation

   1. associate-*r/ 28.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

   2. distribute-lft-in 28.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

   3. metadata-eval 28.0%

      \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

   4. mul-1-neg 28.0%

      \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

5. Simplified 28.0%

   \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

6. Taylor expanded in x around 0 10.4%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]

7. Final simplification 10.4%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023322 
(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))))))