Average Error: 14.0 → 0.2
Time: 11.5s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -953783.0737702926:\\ \;\;\;\;\left(\left(\frac{\frac{x}{F}}{F \cdot \sin B} - \frac{\frac{-1}{F}}{F \cdot \sin B}\right) - \frac{\frac{F}{F}}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 338046798.85707855:\\ \;\;\;\;\frac{\left(F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -953783.0737702926:\\
\;\;\;\;\left(\left(\frac{\frac{x}{F}}{F \cdot \sin B} - \frac{\frac{-1}{F}}{F \cdot \sin B}\right) - \frac{\frac{F}{F}}{\sin B}\right) - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 338046798.85707855:\\
\;\;\;\;\frac{\left(F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}}{\sin B} - \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 -953783.0737702926)
   (-
    (-
     (- (/ (/ x F) (* F (sin B))) (/ (/ -1.0 F) (* F (sin B))))
     (/ (/ F F) (sin B)))
    (/ x (tan B)))
   (if (<= F 338046798.85707855)
     (-
      (/
       (*
        (* F (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.25))
        (sqrt (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5)))
       (sin B))
      (/ 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 <= -953783.0737702926) {
		tmp = ((((x / F) / (F * sin(B))) - ((-1.0 / F) / (F * sin(B)))) - ((F / F) / sin(B))) - (x / tan(B));
	} else if (F <= 338046798.85707855) {
		tmp = (((F * pow(((x * 2.0) + (2.0 + (F * F))), -0.25)) * sqrt(pow(((x * 2.0) + (2.0 + (F * F))), -0.5))) / sin(B)) - (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 < -953783.07377029257

    1. Initial program 25.0

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

    2. Simplified25.0

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

    4. Applied add-sqr-sqrt_binary6425.0

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

    5. Applied associate-*r*_binary6425.0

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

    6. Simplified25.0

      \[\leadsto \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right)} \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\]

    7. Taylor expanded around -inf 3.5

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

    8. Simplified0.2

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

    if -953783.07377029257 < F < 338046798.8570786

    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_binary640.5

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

    5. Applied associate-*r*_binary640.5

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

    6. Simplified0.3

      \[\leadsto \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right)} \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\]
    7. Using strategy rm

    8. Applied associate-*l/_binary640.3

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}}{\sin B}} \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\]

    9. Applied associate-*l/_binary640.3

      \[\leadsto \color{blue}{\frac{\left(F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}}{\sin B}} - \frac{x}{\tan B}\]

    if 338046798.8570786 < 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

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

    3. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -953783.0737702926:\\ \;\;\;\;\left(\left(\frac{\frac{x}{F}}{F \cdot \sin B} - \frac{\frac{-1}{F}}{F \cdot \sin B}\right) - \frac{\frac{F}{F}}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 338046798.85707855:\\ \;\;\;\;\frac{\left(F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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))))))