Average Error: 13.8 → 0.2
Time: 13.8s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -1.794262208773046 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 858337087626449.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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}
\mathbf{if}\;F \leq -1.794262208773046 \cdot 10^{+18}:\\
\;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 858337087626449.4:\\
\;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\

\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
 (if (<= F -1.794262208773046e+18)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 858337087626449.4)
     (- (/ F (* (sin B) (sqrt (+ (* F F) (+ 2.0 (* x 2.0)))))) (/ x (tan B)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))
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 <= -1.794262208773046e+18) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 858337087626449.4) {
		tmp = (F / (sin(B) * sqrt((F * F) + (2.0 + (x * 2.0))))) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if F < -1794262208773046020

    1. Initial program 26.1

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

    2. Simplified26.1

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

    3. Taylor expanded around -inf 0.2

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

    if -1794262208773046020 < F < 858337087626449.375

    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

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

    4. Applied add-sqr-sqrt_binary640.3

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

    5. Applied unpow-prod-down_binary640.3

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

    6. Simplified0.3

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

    7. Simplified0.3

      \[\leadsto \frac{F}{\sin B} \cdot \left({\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5} \cdot \color{blue}{{\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5}}\right) - \frac{x}{\tan B}\]
    8. Using strategy rm

    9. Applied associate-*l/_binary640.3

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

    10. Simplified0.3

      \[\leadsto \frac{\color{blue}{F \cdot \frac{1}{\sqrt{F \cdot F + \left(2 + 2 \cdot x\right)}}}}{\sin B} - \frac{x}{\tan B}\]
    11. Using strategy rm

    12. Applied un-div-inv_binary640.3

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

    13. Applied associate-/l/_binary640.3

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

    if 858337087626449.375 < F

    1. Initial program 27.0

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

    2. Simplified26.9

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

    3. Taylor expanded around inf 0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.794262208773046 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 858337087626449.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))))