VandenBroeck and Keller, Equation (23)

Average Accuracy: 70.0% → 89.9%

Time: 31.1s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

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))))))

↓

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.4e+81)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5000000.0)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* F F)))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_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)));
}

↓

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.4e+81) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5000000.0) {
		tmp = (F * (sqrt((1.0 / (2.0 + (F * F)))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

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

↓

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 <= (-4.4d+81)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5000000.0d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (f * f)))) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4.4e+81) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5000000.0) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (F * F)))) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

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)))

↓

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4.4e+81:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5000000.0:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (F * F)))) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

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

↓

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.4e+81)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
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

↓

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -4.4e+81)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5000000.0)
		tmp = (F * (sqrt((1.0 / (2.0 + (F * F)))) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
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]

↓

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.4e+81], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.39999999999999974e81

   1. Initial program 43.3%

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

   2. Simplified 52.4%

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

      [Start] 43.3	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 43.3	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 43.3	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 52.4	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 52.3	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 52.3	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 87.2%

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

   if -4.39999999999999974e81 < F < 5e6

   1. Initial program 90.6%

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

   2. Simplified 91.3%

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

      [Start] 90.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 90.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 90.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 91.1	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 91.1	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 91.1	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Applied egg-rr 91.2%

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

      [Start] 91.3	\[ F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      add-sqr-sqrt [=>] 91.2	\[ F \cdot \frac{{\color{blue}{\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      unpow-prod-down [=>] 91.2	\[ F \cdot \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]

   4. Simplified 91.2%

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

      [Start] 91.2	\[ F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      pow-sqr [=>] 91.2	\[ F \cdot \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(2 \cdot -0.5\right)}}}{\sin B} - \frac{x}{\tan B} \]
      metadata-eval [=>] 91.2	\[ F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-1}}}{\sin B} - \frac{x}{\tan B} \]
      unpow-1 [=>] 91.2	\[ F \cdot \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B} - \frac{x}{\tan B} \]
      fma-udef [=>] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\color{blue}{x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]
      *-commutative [<=] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B} \]
      fma-udef [=>] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{2 \cdot x + \color{blue}{\left(F \cdot F + 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [<=] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{2 \cdot x + \left(\color{blue}{{F}^{2}} + 2\right)}}}{\sin B} - \frac{x}{\tan B} \]
      +-commutative [<=] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\color{blue}{\left({F}^{2} + 2\right) + 2 \cdot x}}}}{\sin B} - \frac{x}{\tan B} \]
      associate-+r+ [<=] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}}{\sin B} - \frac{x}{\tan B} \]
      fma-def [=>] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}}{\sin B} - \frac{x}{\tan B} \]
      +-commutative [<=] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}}{\sin B} - \frac{x}{\tan B} \]
      fma-def [=>] 91.2	\[ F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}}{\sin B} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 91.2%

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

   6. Simplified 91.2%

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

      [Start] 91.2	\[ F \cdot \frac{\sqrt{\frac{1}{{F}^{2} + 2}}}{\sin B} - \frac{x}{\tan B} \]
      +-commutative [=>] 91.2	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 91.2	\[ F \cdot \frac{\sqrt{\frac{1}{2 + \color{blue}{F \cdot F}}}}{\sin B} - \frac{x}{\tan B} \]

   if 5e6 < F

   1. Initial program 52.6%

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

   2. Simplified 61.2%

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

      [Start] 52.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 52.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 52.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 61.1	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 61.1	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 61.1	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 89.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.4%
Cost	20424

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Accuracy	89.4%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Accuracy	89.4%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Accuracy	80.0%
Cost	14608

\[\begin{array}{l} t_0 := 2 + F \cdot F\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\frac{1}{t_0}} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 115000:\\ \;\;\;\;\left(F \cdot t_1\right) \cdot {\left(x \cdot 2 + t_0\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_2\\ \end{array} \]

Alternative 5
Accuracy	80.0%
Cost	14480

\[\begin{array}{l} t_0 := 2 + F \cdot F\\ t_1 := \frac{F}{\sin B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\frac{1}{t_0}} \cdot t_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1650000:\\ \;\;\;\;t_1 \cdot {\left(x \cdot 2 + t_0\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \]

Alternative 6
Accuracy	80.0%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 145000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 7
Accuracy	79.8%
Cost	14160

\[\begin{array}{l} t_0 := \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0255:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.07:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 8
Accuracy	75.7%
Cost	13776

\[\begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.000102:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.00375:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 9
Accuracy	55.4%
Cost	13648

\[\begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{if}\;F \leq -24000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 0.14:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 10
Accuracy	61.9%
Cost	13648

\[\begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{if}\;F \leq -24000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq 0.088:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 11
Accuracy	61.9%
Cost	13648

\[\begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{if}\;F \leq -24000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 0.27:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 12
Accuracy	62.0%
Cost	13648

\[\begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{if}\;F \leq -24000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.108:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Accuracy	69.0%
Cost	13648

\[\begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{if}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Accuracy	53.9%
Cost	7816

\[\begin{array}{l} \mathbf{if}\;F \leq 2.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.94:\\ \;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 15
Accuracy	53.8%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.075:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 16
Accuracy	47.7%
Cost	7245

\[\begin{array}{l} \mathbf{if}\;F \leq 2.15 \cdot 10^{-19} \lor \neg \left(F \leq 4.3 \cdot 10^{+182}\right) \land F \leq 5.4 \cdot 10^{+224}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 17
Accuracy	51.2%
Cost	7245

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{-190}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{+248} \lor \neg \left(F \leq 1.45 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Accuracy	52.0%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;F \leq 1.52 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 19
Accuracy	41.6%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 20
Accuracy	37.6%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Accuracy	33.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Accuracy	26.8%
Cost	716

\[\begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -2.6 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+222}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5.9 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 23
Accuracy	24.2%
Cost	653

\[\begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+222} \lor \neg \left(F \leq 1.22 \cdot 10^{+275}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 24
Accuracy	33.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 25
Accuracy	18.1%
Cost	324

\[\begin{array}{l} \mathbf{if}\;F \leq 4.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 26
Accuracy	10.6%
Cost	192

\[\frac{-1}{B} \]

Reproduce

herbie shell --seed 2023153 
(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))))))