Average Error: 13.5 → 0.2
Time: 18.2s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -21728531036498.848:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 24953040.116247203:\\ \;\;\;\;F \cdot \frac{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -21728531036498.848:\\
\;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 24953040.116247203:\\
\;\;\;\;F \cdot \frac{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\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 -21728531036498.848)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 24953040.116247203)
     (-
      (* F (/ (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5) (sin B)))
      (/ x (tan B)))
     (- (/ (- 1.0 (+ (/ x (* F F)) (/ 1.0 (* F F)))) (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 <= -21728531036498.848) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 24953040.116247203) {
		tmp = (F * (pow(((x * 2.0) + (2.0 + (F * F))), -0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = ((1.0 - ((x / (F * F)) + (1.0 / (F * F)))) / 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 < -21728531036498.8477

    1. Initial program 26.3

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

    2. Simplified26.2

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

    3. Taylor expanded around -inf 0.1

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

    if -21728531036498.8477 < F < 24953040.1162472032

    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 div-inv_binary640.3

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

    5. Applied associate-*l*_binary640.3

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

    6. Simplified0.3

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

    if 24953040.1162472032 < F

    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. Simplified24.1

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

    4. Applied associate-*l/_binary6418.5

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

    5. Simplified18.5

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

    6. Taylor expanded around inf 0.2

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

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -21728531036498.848:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 24953040.116247203:\\ \;\;\;\;F \cdot \frac{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021097 
(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))))))