VandenBroeck and Keller, Equation (23)

Average Error: 14.1 → 0.4

Time: 28.4s

Precision: binary64

Cost: 27144

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.95 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

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 (/ 1.0 (tan B)))))
   (if (<= F -3.95e+46)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5e+34)
       (-
        (* (/ F (sin B)) (pow (+ x (+ (+ (* F F) 2.0) x)) -0.5))
        (/ (* (cos B) x) (sin B)))
       (- (/ 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 * (1.0 / tan(B));
	double tmp;
	if (F <= -3.95e+46) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5e+34) {
		tmp = ((F / sin(B)) * pow((x + (((F * F) + 2.0) + x)), -0.5)) - ((cos(B) * x) / sin(B));
	} 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 * (1.0d0 / tan(b))
    if (f <= (-3.95d+46)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5d+34) then
        tmp = ((f / sin(b)) * ((x + (((f * f) + 2.0d0) + x)) ** (-0.5d0))) - ((cos(b) * x) / sin(b))
    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 * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -3.95e+46) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5e+34) {
		tmp = ((F / Math.sin(B)) * Math.pow((x + (((F * F) + 2.0) + x)), -0.5)) - ((Math.cos(B) * x) / Math.sin(B));
	} 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 * (1.0 / math.tan(B))
	tmp = 0
	if F <= -3.95e+46:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5e+34:
		tmp = ((F / math.sin(B)) * math.pow((x + (((F * F) + 2.0) + x)), -0.5)) - ((math.cos(B) * x) / math.sin(B))
	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 * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -3.95e+46)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5e+34)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(x + Float64(Float64(Float64(F * F) + 2.0) + x)) ^ -0.5)) - Float64(Float64(cos(B) * x) / sin(B)));
	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 * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -3.95e+46)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5e+34)
		tmp = ((F / sin(B)) * ((x + (((F * F) + 2.0) + x)) ^ -0.5)) - ((cos(B) * x) / sin(B));
	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[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.95e+46], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5e+34], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x + N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9500000000000002e46

   1. Initial program 29.6

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

   2. Simplified 29.6

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

      [Start] 29.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 29.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 29.6	\[ \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 29.6	\[ \color{blue}{\left(0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) - x \cdot \frac{1}{\tan B}} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 29.6	\[ \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + 0\right)} - x \cdot \frac{1}{\tan B} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 29.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 0.2

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

   if -3.9500000000000002e46 < F < 4.9999999999999998e34

   1. Initial program 0.6

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

   2. Simplified 0.6

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

      [Start] 0.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 0.6	\[ \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 0.6	\[ \color{blue}{\left(0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) - x \cdot \frac{1}{\tan B}} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.6	\[ \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + 0\right)} - x \cdot \frac{1}{\tan B} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in x around 0 0.5

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

   if 4.9999999999999998e34 < F

   1. Initial program 26.9

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

   2. Simplified 26.9

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

      [Start] 26.9	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 26.9	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 26.9	\[ \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 26.9	\[ \color{blue}{\left(0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) - x \cdot \frac{1}{\tan B}} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 26.9	\[ \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + 0\right)} - x \cdot \frac{1}{\tan B} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 26.9	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.95 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	20744

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.92 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 4000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Error	1.4
Cost	20552

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Error	1.4
Cost	20168

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Error	5.5
Cost	14480

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{x}{B}\\ t_1 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.05 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{B} - t_1\\ \mathbf{elif}\;F \leq 160000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 5
Error	6.4
Cost	14348

\[\begin{array}{l} t_0 := {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5}\\ t_1 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.92 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{F}{B} \cdot t_0 - t_1\\ \mathbf{elif}\;F \leq 300000:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 6
Error	6.6
Cost	14152

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.075:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Error	8.2
Cost	13904

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5} \cdot F}{\sin B} - \frac{x}{B}\\ t_1 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-165}:\\ \;\;\;\;-\frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 0.125:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 8
Error	11.2
Cost	13772

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -8.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;-\frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 9
Error	6.6
Cost	13768

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.075:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 10
Error	20.8
Cost	13580

\[\begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 11
Error	15.8
Cost	13580

\[\begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -2.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 12
Error	26.7
Cost	8200

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ t_1 := \frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-125}:\\ \;\;\;\;\left(\frac{F}{B} + F \cdot \left(B \cdot 0.16666666666666666\right)\right) \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-292}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Error	27.1
Cost	7892

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ t_1 := \frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-292}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Error	27.0
Cost	7892

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ t_1 := \frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 9.7 \cdot 10^{-293}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 15
Error	26.7
Cost	7892

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ t_1 := \frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-94}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(x + \left(\left(F \cdot F + 2\right) + x\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-293}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 16
Error	27.0
Cost	7628

\[\begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-292}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17
Error	27.3
Cost	7372

\[\begin{array}{l} t_0 := \frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.4 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-292}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 18
Error	28.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 19
Error	35.9
Cost	6988

\[\begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-55}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 20
Error	31.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-24}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 21
Error	35.3
Cost	6920

\[\begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-24}:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 22
Error	38.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-22}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]

Alternative 23
Error	39.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-21}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]

Alternative 24
Error	39.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-34}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 25
Error	44.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -9.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 26
Error	47.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 27
Error	56.9
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))))