VandenBroeck and Keller, Equation (23)

Average Error: 13.5 → 0.2

Time: 50.5s

Precision: binary64

Cost: 39496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

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 -7.5e+14)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 47000000.0)
     (fma (/ F (sin B)) (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ (- x) (tan B)))
     (- (/ 1.0 (sin B)) (/ x (/ (sin B) (cos 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 <= -7.5e+14) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 47000000.0) {
		tmp = fma((F / sin(B)), pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - (x / (sin(B) / cos(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 <= -7.5e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 47000000.0)
		tmp = fma(Float64(F / sin(B)), (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / Float64(sin(B) / cos(B))));
	end
	return 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, -7.5e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 47000000.0], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.5e14

   1. Initial program 25.4

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

   2. Simplified 25.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
      Proof
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (fma.f64 x 2 (fma.f64 F F 2)) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (fma.f64 x 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 F F) 2))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 2) (+.f64 (*.f64 F F) 2))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 x)) (+.f64 (*.f64 F F) 2)) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (Rewrite<= metadata-eval (neg.f64 1/2))) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 x) 1)) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 x) (/.f64 1 (tan.f64 B))))): 31 points increase in error, 10 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2)))) (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 2 points increase in error, 1 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in F around -inf 13.3

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

   4. Applied egg-rr 0.2

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

   5. Taylor expanded in F around 0 0.1

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

   if -7.5e14 < F < 4.7e7

   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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
      Proof
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (fma.f64 x 2 (fma.f64 F F 2)) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (fma.f64 x 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 F F) 2))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 2) (+.f64 (*.f64 F F) 2))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 x)) (+.f64 (*.f64 F F) 2)) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (Rewrite<= metadata-eval (neg.f64 1/2))) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 x) 1)) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 x) (/.f64 1 (tan.f64 B))))): 31 points increase in error, 10 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2)))) (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 2 points increase in error, 1 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2)))))): 0 points increase in error, 0 points decrease in error

   if 4.7e7 < F

   1. Initial program 24.5

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

   2. Simplified 24.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
      Proof
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (fma.f64 x 2 (fma.f64 F F 2)) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (fma.f64 x 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 F F) 2))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 2) (+.f64 (*.f64 F F) 2))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 x)) (+.f64 (*.f64 F F) 2)) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x))) -1/2) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (Rewrite<= metadata-eval (neg.f64 1/2))) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) (/.f64 (neg.f64 x) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 x) 1)) (tan.f64 B))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 x) (/.f64 1 (tan.f64 B))))): 31 points increase in error, 10 points decrease in error
      (fma.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2))) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2)))) (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))))): 2 points increase in error, 1 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (*.f64 x (/.f64 1 (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) 2) (*.f64 2 x)) (neg.f64 (/.f64 1 2)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in F around inf 0.2

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

   4. Simplified 0.2

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}} \]
      Proof
      (-.f64 (/.f64 1 (sin.f64 B)) (/.f64 x (/.f64 (sin.f64 B) (cos.f64 B)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 1 (sin.f64 B)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (cos.f64 B)) (sin.f64 B)))): 7 points increase in error, 12 points decrease in error
      (-.f64 (/.f64 1 (sin.f64 B)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 B) x)) (sin.f64 B))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 1 (sin.f64 B)) (neg.f64 (/.f64 (*.f64 (cos.f64 B) x) (sin.f64 B))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (sin.f64 B)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (cos.f64 B) x) (sin.f64 B))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20744

\[\begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	20552

\[\begin{array}{l} \mathbf{if}\;F \leq -240:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	20552

\[\begin{array}{l} \mathbf{if}\;F \leq -240:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

Alternative 4
Error	7.9
Cost	20304

\[\begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

Alternative 5
Error	0.7
Cost	20168

\[\begin{array}{l} \mathbf{if}\;F \leq -240:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \end{array} \]

Alternative 6
Error	7.9
Cost	14480

\[\begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 7
Error	7.9
Cost	14288

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -7.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 8
Error	10.5
Cost	13772

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -12500000:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -12500000:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 11
Error	22.7
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -12500000:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 12
Error	17.9
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 13
Error	28.8
Cost	7888

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-256}:\\ \;\;\;\;\frac{F \cdot t_0 - x}{B}\\ \mathbf{elif}\;F \leq 10^{-156}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{-1}{F}}{\frac{B}{F}}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 14
Error	28.8
Cost	7760

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 10^{-156}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{-1}{F}}{\frac{B}{F}}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Error	31.3
Cost	7640

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -4.292671720788959 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.800367105507675 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq -1.12906808814594 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.665335503308794 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \leq 2.4554554563438134 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.937676715662612 \cdot 10^{-12}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	31.3
Cost	7640

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;x \leq -4.292671720788959 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.800367105507675 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq -1.12906808814594 \cdot 10^{-223}:\\ \;\;\;\;-1 + \left(t_1 + 1\right)\\ \mathbf{elif}\;x \leq 8.665335503308794 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \leq 2.4554554563438134 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.937676715662612 \cdot 10^{-12}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	34.8
Cost	7116

\[\begin{array}{l} \mathbf{if}\;F \leq -35000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Error	35.4
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -35000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-39}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Error	37.3
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -35000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Error	39.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -5.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Error	45.2
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -126000000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-39}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 22
Error	42.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -5.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-39}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 23
Error	52.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;F \leq 6.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 24
Error	57.3
Cost	192

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

Reproduce

herbie shell --seed 2022325 
(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))))))