Average Error: 13.8 → 0.3
Time: 11.5s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -6391356515907.039:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 111746443.95311779:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B \cdot \left(1 + \frac{x}{F \cdot F}\right)} - \frac{x}{\tan B}\\ \end{array}\]
\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 -6391356515907.039:\\
\;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 111746443.95311779:\\
\;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{1}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin B \cdot \left(1 + \frac{x}{F \cdot F}\right)} - \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 -6391356515907.039)
   (- (/ (+ -1.0 (/ 1.0 (* F F))) (sin B)) (/ x (tan B)))
   (if (<= F 111746443.95311779)
     (-
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
      (* x (/ 1.0 (tan B))))
     (- (/ 1.0 (* (sin B) (+ 1.0 (/ x (* F F))))) (/ 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 <= -6391356515907.039) {
		tmp = ((-1.0 + (1.0 / (F * F))) / sin(B)) - (x / tan(B));
	} else if (F <= 111746443.95311779) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x * (1.0 / tan(B)));
	} else {
		tmp = (1.0 / (sin(B) * (1.0 + (x / (F * F))))) - (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 < -6391356515907.0391

    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. Using strategy rm

    4. Applied associate-*l/_binary6420.4

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

    5. Simplified20.4

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

    6. Taylor expanded around -inf 0.2

      \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} - 1}}{\sin B} - \frac{x}{\tan B}\]

    7. Simplified0.2

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

    if -6391356515907.0391 < F < 111746443.95311779

    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_binary640.4

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

    if 111746443.95311779 < F

    1. Initial program 24.6

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

    2. Simplified24.5

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

    4. Applied associate-*l/_binary6418.6

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

    5. Simplified18.6

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

    7. Applied clear-num_binary6418.7

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

    8. Taylor expanded around inf 0.1

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

    9. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6391356515907.039:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 111746443.95311779:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B \cdot \left(1 + \frac{x}{F \cdot F}\right)} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(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))))))