VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 0.6

Time: 49.3s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8960439192206498 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 8.085133053421346 \cdot 10^{+36}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

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
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.8960439192206498e+93)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 8.085133053421346e+36)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 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) {
	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 <= -3.8960439192206498e+93) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 8.085133053421346e+36) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (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
    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 <= (-3.8960439192206498d+93)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 8.085133053421346d+36) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (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) {
	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 <= -3.8960439192206498e+93) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 8.085133053421346e+36) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (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):
	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 <= -3.8960439192206498e+93:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 8.085133053421346e+36:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (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)
	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 <= -3.8960439192206498e+93)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 8.085133053421346e+36)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + 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 = 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 <= -3.8960439192206498e+93)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 8.085133053421346e+36)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (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_] := 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, -3.8960439192206498e+93], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8.085133053421346e+36], 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] - 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 < -3.8960439192206498e93

   1. Initial program 33.5

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

   2. Applied egg-rr 26.9

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

   3. Taylor expanded in F around -inf 0.2

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

   4. Applied egg-rr 0.2

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

   if -3.8960439192206498e93 < F < 8.08513305342134575e36

   1. Initial program 1.0

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

   2. Applied egg-rr 0.9

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

   if 8.08513305342134575e36 < F

   1. Initial program 28.3

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

   2. Applied egg-rr 21.7

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

   3. Taylor expanded in F around inf 0.2

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8960439192206498 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.085133053421346 \cdot 10^{+36}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	20168

\[\begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-5}:\\ \;\;\;\;t_0 + \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2
Error	5.9
Cost	14420

\[\begin{array}{l} t_0 := \sqrt{2 + x \cdot 2}\\ t_1 := \frac{F}{\sin B \cdot t_0} - \frac{x}{B}\\ t_2 := x \cdot \frac{-1}{\tan B}\\ t_3 := t_2 + \frac{F}{B \cdot t_0}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 10^{-103}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 3
Error	5.8
Cost	14420

\[\begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \sqrt{2 + x \cdot 2}\\ t_2 := t_0 + \frac{F}{B \cdot t_1}\\ \mathbf{if}\;F \leq -340:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 10^{-70}:\\ \;\;\;\;\frac{F}{\sin B \cdot t_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4
Error	7.8
Cost	14160

\[\begin{array}{l} t_0 := \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.85 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.025:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5
Error	24.1
Cost	14108

\[\begin{array}{l} t_0 := x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{if}\;F \leq -340:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.04 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 0.025:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.928824732976153 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	24.1
Cost	14108

\[\begin{array}{l} t_0 := -\cos B\\ t_1 := \frac{x \cdot t_0}{\sin B}\\ \mathbf{if}\;F \leq -340:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.04 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{t_0}{\sin B}\\ \mathbf{elif}\;F \leq 0.025:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.928824732976153 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.9
Cost	14108

\[\begin{array}{l} t_0 := -\cos B\\ t_1 := \frac{x \cdot t_0}{\sin B}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.04 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{t_0}{\sin B}\\ \mathbf{elif}\;F \leq 0.025:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.928824732976153 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	9.9
Cost	13904

\[\begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.04 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9
Error	28.1
Cost	13648

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -3.388011687540596 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.3395459180828454 \cdot 10^{-57}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;x \leq -4.784891647918743 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9845655496865334 \cdot 10^{-150}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;x \leq 2.142598733722326 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	31.2
Cost	8152

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{F}{B} \cdot t_0 - \frac{x}{B}\\ t_2 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -3.388011687540596 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.3395459180828454 \cdot 10^{-57}:\\ \;\;\;\;\frac{F \cdot t_0 - x}{B}\\ \mathbf{elif}\;x \leq -4.784891647918743 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.926440180854666 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.100770164993991 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 265.88939426725454:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	31.2
Cost	8024

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -3.388011687540596 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3395459180828454 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.784891647918743 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.926440180854666 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.100770164993991 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 265.88939426725454:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	30.5
Cost	7244

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -8.777406247443913 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.987687951265954 \cdot 10^{-267}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 2.142598733722326 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	38.3
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 14
Error	35.7
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -4.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Error	49.4
Cost	584

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -1.3395459180828454 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.260773538053991 \cdot 10^{-268}:\\ \;\;\;\;-1 + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	40.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 10^{-60}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 17
Error	49.8
Cost	520

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -5.443389454077265 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.987687951265954 \cdot 10^{-267}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	45.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 19
Error	56.8
Cost	192

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

Reproduce

herbie shell --seed 2022291 
(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))))))