VandenBroeck and Keller, Equation (23)

Average Error: 13.3 → 0.2

Time: 32.5s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -48000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 390000000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

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 -48000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 390000000000.0)
       (- (* (/ F (sin B)) (pow (+ (* F F) (+ 2.0 (+ x x))) -0.5)) 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 <= -48000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 390000000000.0) {
		tmp = ((F / sin(B)) * pow(((F * F) + (2.0 + (x + x))), -0.5)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    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 <= (-48000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 390000000000.0d0) then
        tmp = ((f / sin(b)) * (((f * f) + (2.0d0 + (x + x))) ** (-0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	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 <= -48000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 390000000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((F * F) + (2.0 + (x + x))), -0.5)) - 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 <= -48000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 390000000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((F * F) + (2.0 + (x + x))), -0.5)) - 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 <= -48000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 390000000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(F * F) + Float64(2.0 + Float64(x + x))) ^ -0.5)) - 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 <= -48000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 390000000000.0)
		tmp = ((F / sin(B)) * (((F * F) + (2.0 + (x + x))) ^ -0.5)) - 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, -48000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 390000000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.8e10

   1. Initial program 25.1

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

   2. Simplified 25.0

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

      [Start] 25.1	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      rational.json-simplify-1 [=>] 25.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      rational.json-simplify-36 [=>] 25.1	\[ \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
      rational.json-simplify-12 [=>] 25.1	\[ \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + 0\right) - x \cdot \frac{1}{\tan B}} \]
      rational.json-simplify-8 [=>] 25.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 0.2

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

   if -4.8e10 < F < 3.9e11

   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. Simplified 0.3

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

      [Start] 0.4	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      rational.json-simplify-36 [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
      rational.json-simplify-12 [=>] 0.4	\[ \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + 0\right) - x \cdot \frac{1}{\tan B}} \]
      rational.json-simplify-8 [=>] 0.4	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} - x \cdot \frac{1}{\tan B} \]

   if 3.9e11 < F

   1. Initial program 24.7

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

   2. Simplified 24.7

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

      [Start] 24.7	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      rational.json-simplify-1 [=>] 24.7	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      rational.json-simplify-36 [=>] 24.7	\[ \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
      rational.json-simplify-12 [=>] 24.7	\[ \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + 0\right) - x \cdot \frac{1}{\tan B}} \]
      rational.json-simplify-8 [=>] 24.7	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -48000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 390000000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	20424

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

Alternative 2
Error	0.6
Cost	20296

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

Alternative 3
Error	5.8
Cost	14216

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -38000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 170000:\\ \;\;\;\;\frac{F}{B} \cdot {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Error	6.0
Cost	14024

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.88:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.36:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Error	11.2
Cost	13776

\[\begin{array}{l} t_0 := \frac{F}{B} \cdot {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -225:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2550:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 6
Error	21.1
Cost	13580

\[\begin{array}{l} t_0 := {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -68000000000:\\ \;\;\;\;\frac{-1}{B} - t_1\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-177}:\\ \;\;\;\;\left(\frac{1}{B} \cdot F\right) \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-174}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 8000:\\ \;\;\;\;\frac{F}{B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_1\\ \end{array} \]

Alternative 7
Error	15.8
Cost	13580

\[\begin{array}{l} t_0 := \frac{F}{B} \cdot {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.8:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 22000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_1\\ \end{array} \]

Alternative 8
Error	25.5
Cost	7816

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -45000000000:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 27500:\\ \;\;\;\;\frac{F}{B} \cdot {\left(F \cdot F + \left(2 + \left(x + x\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 9
Error	25.6
Cost	7688

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0085:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 0.36:\\ \;\;\;\;\frac{-x}{B} + \sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 10
Error	29.3
Cost	7640

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{B} - t_0\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{+180}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	28.5
Cost	7500

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-229}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-35}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - t_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+184}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 12
Error	25.6
Cost	7496

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+184}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 13
Error	28.6
Cost	7376

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{B} - t_0\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	30.6
Cost	7244

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-209}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Error	34.4
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 16
Error	37.6
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -2.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \end{array} \]

Alternative 17
Error	40.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \end{array} \]

Alternative 18
Error	40.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 19
Error	45.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]

Alternative 20
Error	50.2
Cost	256

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

Reproduce

herbie shell --seed 2023073 
(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))))))