Average Error: 13.5 → 0.2
Time: 12.5s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -4.62881705503341 \cdot 10^{+116}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 2079208.570211153:\\ \;\;\;\;\frac{F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot \left(F \cdot F\right)} + \frac{1}{\sin B \cdot \left(F \cdot F\right)}\right)\right) - \frac{x}{\tan B}\\ \end{array}\]
\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 -4.62881705503341 \cdot 10^{+116}:\\
\;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\

\mathbf{elif}\;F \leq 2079208.570211153:\\
\;\;\;\;\frac{F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot \left(F \cdot F\right)} + \frac{1}{\sin B \cdot \left(F \cdot F\right)}\right)\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 -4.62881705503341e+116)
   (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
   (if (<= F 2079208.570211153)
     (-
      (/ (* F (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5)) (sin B))
      (/ x (tan B)))
     (-
      (-
       (/ 1.0 (sin B))
       (+ (/ x (* (sin B) (* F F))) (/ 1.0 (* (sin B) (* 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 <= -4.62881705503341e+116) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 2079208.570211153) {
		tmp = ((F * pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) / sin(B)) - (x / tan(B));
	} else {
		tmp = ((1.0 / sin(B)) - ((x / (sin(B) * (F * F))) + (1.0 / (sin(B) * (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 < -4.62881705503341003e116

    1. Initial program 35.3

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

    2. Taylor expanded around -inf 0.2

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

    if -4.62881705503341003e116 < F < 2079208.570211153

    1. Initial program 1.0

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

    2. Simplified0.9

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

    4. Applied associate-*l/_binary640.3

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

    5. Simplified0.3

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

    if 2079208.570211153 < F

    1. Initial program 24.9

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

    2. Simplified24.9

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.62881705503341 \cdot 10^{+116}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 2079208.570211153:\\ \;\;\;\;\frac{F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot \left(F \cdot F\right)} + \frac{1}{\sin B \cdot \left(F \cdot F\right)}\right)\right) - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021148 
(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))))))