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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -84561923.8329791:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 199346.12227644376:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}}{\sin B} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sin B \cdot {F}^{2}\\ \left(\frac{1}{\sin B} - \left(\frac{x}{t_1} + \frac{1}{t_1}\right)\right) - t_0 \end{array}\\ \end{array} \]

\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 -84561923.8329791:\\
\;\;\;\;\frac{-1}{\sin B} - t_0\\

\mathbf{elif}\;F \leq 199346.12227644376:\\
\;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}}{\sin B} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - t_0\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \sin B \cdot {F}^{2}\\
\left(\frac{1}{\sin B} - \left(\frac{x}{t_1} + \frac{1}{t_1}\right)\right) - t_0
\end{array}\\


\end{array}

(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 -84561923.8329791)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 199346.12227644376)
       (-
        (*
         (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.25)) (sin B))
         (sqrt (pow (fma x 2.0 (fma F F 2.0)) -0.5)))
        t_0)
       (let* ((t_1 (* (sin B) (pow F 2.0))))
         (- (- (/ 1.0 (sin B)) (+ (/ x t_1) (/ 1.0 t_1))) t_0))))))

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 <= -84561923.8329791) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 199346.12227644376) {
		tmp = (((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.25)) / sin(B)) * sqrt(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5))) - t_0;
	} else {
		double t_1 = sin(B) * pow(F, 2.0);
		tmp = ((1.0 / sin(B)) - ((x / t_1) + (1.0 / t_1))) - t_0;
	}
	return tmp;
}

Derivation

Split input into 3 regimes
if F < -84561923.832979098
1. Initial program 24.2
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Simplified24.2
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
3. Taylor expanded in F around -inf 0.1
  \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]
if -84561923.832979098 < F < 199346.12227644376
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. Simplified0.3
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
3. Applied add-sqr-sqrt_binary640.6
  \[\leadsto \frac{F}{\sin B} \cdot \color{blue}{\left(\sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}\right)} - \frac{x}{\tan B} \]
4. Applied associate-*r*_binary640.5
  \[\leadsto \color{blue}{\left(\frac{F}{\sin B} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}\right) \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - \frac{x}{\tan B} \]
5. Simplified0.3
  \[\leadsto \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right)} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - \frac{x}{\tan B} \]
6. Applied associate-*l/_binary640.3
  \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}}{\sin B}} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - \frac{x}{\tan B} \]
if 199346.12227644376 < F
1. Initial program 26.0
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Simplified26.0
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
3. Taylor expanded in F around inf 0.2
  \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot {F}^{2}} + \frac{1}{\sin B \cdot {F}^{2}}\right)\right)} - \frac{x}{\tan B} \]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -84561923.8329791:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 199346.12227644376:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}}{\sin B} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot {F}^{2}} + \frac{1}{\sin B \cdot {F}^{2}}\right)\right) - \frac{x}{\tan B}\\ \end{array} \]

Error

Derivation

`if F < -84561923.832979098`

`if -84561923.832979098 < F < 199346.12227644376`

`if 199346.12227644376 < F`

Reproduce