VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 0.3

Time: 1.2min

Precision: binary64

Cost: 27144

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -340000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 28.5:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

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
 (if (<= F -340000000.0)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 (sin B)))
   (if (<= F 28.5)
     (-
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
      (* (/ x (sin B)) (cos B)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

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 tmp;
	if (F <= -340000000.0) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	} else if (F <= 28.5) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x / sin(B)) * cos(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    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) :: tmp
    if (f <= (-340000000.0d0)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / sin(b))
    else if (f <= 28.5d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - ((x / sin(b)) * cos(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	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 tmp;
	if (F <= -340000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / Math.sin(B));
	} else if (F <= 28.5) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x / Math.sin(B)) * Math.cos(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	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):
	tmp = 0
	if F <= -340000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / math.sin(B))
	elif F <= 28.5:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x / math.sin(B)) * math.cos(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	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)
	tmp = 0.0
	if (F <= -340000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / sin(B)));
	elseif (F <= 28.5)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(Float64(x / sin(B)) * cos(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	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)
	tmp = 0.0;
	if (F <= -340000000.0)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	elseif (F <= 28.5)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - ((x / sin(B)) * cos(B));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	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_] := If[LessEqual[F, -340000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 28.5], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.4e8

   1. Initial program 26.5

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

   2. Taylor expanded in F around -inf 0.2

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

   if -3.4e8 < F < 28.5

   1. Initial program 0.4

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.3

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

   if 28.5 < F

   1. Initial program 24.8

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

   2. Applied egg-rr 23.2

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

   3. Taylor expanded in F around inf 0.5

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

   4. Applied egg-rr 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -340000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 28.5:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20616

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -340000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 28.5:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	20424

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -290000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	20168

\[\begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -500000:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10:\\ \;\;\;\;t_0 + F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4
Error	8.1
Cost	14288

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5
Error	8.1
Cost	14288

\[\begin{array}{l} t_0 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-142}:\\ \;\;\;\;t_0 \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-42}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6
Error	11.7
Cost	13908

\[\begin{array}{l} t_0 := -\cos B\\ t_1 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{-75}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-142}:\\ \;\;\;\;t_1 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot t_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{F \cdot t_1 - x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \frac{t_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7
Error	18.2
Cost	13776

\[\begin{array}{l} t_0 := -\cos B\\ \mathbf{if}\;F \leq -800000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot t_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \frac{t_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8
Error	25.5
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -800000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 9
Error	25.5
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -800000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 10
Error	30.4
Cost	8164

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{\frac{F}{-0.5 \cdot \frac{2 + x \cdot 2}{F} - F} - x}{B}\\ t_2 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;x \leq -3.265693301720736 \cdot 10^{-23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.543195578376606 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7078417522561465 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.9531053242878095 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.3107171265862537 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.391977702784811 \cdot 10^{-259}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;x \leq 2.7924663554414065 \cdot 10^{-211}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 5.264213198462519 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.300682684352538 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	30.6
Cost	8164

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ t_2 := 2 + x \cdot 2\\ t_3 := \frac{\frac{F}{-0.5 \cdot \frac{t_2}{F} - F} - x}{B}\\ \mathbf{if}\;x \leq -3.265693301720736 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.543195578376606 \cdot 10^{-134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.7078417522561465 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.9531053242878095 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.3107171265862537 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1471728084220373 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{t_2}} - x}{B}\\ \mathbf{elif}\;x \leq 2.7924663554414065 \cdot 10^{-211}:\\ \;\;\;\;1 + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.264213198462519 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.300682684352538 \cdot 10^{-28}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	30.6
Cost	8164

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ t_2 := 2 + x \cdot 2\\ t_3 := \frac{\frac{F}{-0.5 \cdot \frac{t_2}{F} - F} - x}{B}\\ \mathbf{if}\;x \leq -3.265693301720736 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.543195578376606 \cdot 10^{-134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.7078417522561465 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.9531053242878095 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.3107171265862537 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1471728084220373 \cdot 10^{-243}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_2}} - x}{B}\\ \mathbf{elif}\;x \leq 2.7924663554414065 \cdot 10^{-211}:\\ \;\;\;\;1 + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.264213198462519 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.300682684352538 \cdot 10^{-28}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	35.1
Cost	7248

\[\begin{array}{l} t_0 := \sqrt{0.5} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-132}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 14
Error	35.2
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Error	37.7
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{F}{-0.5 \cdot \frac{2 + x \cdot 2}{F} - F} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 16
Error	40.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}} - x}{B}\\ \end{array} \]

Alternative 17
Error	40.0
Cost	1220

\[\begin{array}{l} t_0 := \frac{2 + x \cdot 2}{F}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{F}{-0.5 \cdot t_0 - F} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot t_0} - x}{B}\\ \end{array} \]

Alternative 18
Error	40.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1 - x}{F \cdot F}\right) - x}{B}\\ \end{array} \]

Alternative 19
Error	40.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Error	45.0
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 21
Error	42.6
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 22
Error	52.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-246}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 23
Error	57.3
Cost	192

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

Reproduce

herbie shell --seed 2022310 
(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))))))