Average Error: 13.8 → 0.3
Time: 11.0s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -6.533745919767842 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 27842.90360465268:\\ \;\;\;\;\left({\left(\frac{1}{F \cdot F + 2}\right)}^{0.25} \cdot \frac{F}{\sin B}\right) \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right)\right) - \frac{x}{\tan B}\\ \end{array}\]
\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 -6.533745919767842 \cdot 10^{+40}:\\
\;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 27842.90360465268:\\
\;\;\;\;\left({\left(\frac{1}{F \cdot F + 2}\right)}^{0.25} \cdot \frac{F}{\sin B}\right) \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\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 -6.533745919767842e+40)
   (-
    (+
     (+ (/ x (* (* F F) (sin B))) (/ 1.0 (* (* F F) (sin B))))
     (/ -1.0 (sin B)))
    (/ x (tan B)))
   (if (<= F 27842.90360465268)
     (-
      (*
       (* (pow (/ 1.0 (+ (* F F) 2.0)) 0.25) (/ F (sin B)))
       (sqrt (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)))
      (/ x (tan B)))
     (-
      (-
       (/ 1.0 (sin B))
       (+ (/ x (* (* F F) (sin B))) (/ 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 <= -6.533745919767842e+40) {
		tmp = (((x / ((F * F) * sin(B))) + (1.0 / ((F * F) * sin(B)))) + (-1.0 / sin(B))) - (x / tan(B));
	} else if (F <= 27842.90360465268) {
		tmp = ((pow((1.0 / ((F * F) + 2.0)), 0.25) * (F / sin(B))) * sqrt(pow((((F * F) + 2.0) + (x * 2.0)), -0.5))) - (x / tan(B));
	} else {
		tmp = ((1.0 / sin(B)) - ((x / ((F * F) * sin(B))) + (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 < -6.53374591976784205e40

    1. Initial program 29.2

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

    2. Simplified29.1

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -6.53374591976784205e40 < F < 27842.903604652682

    1. Initial program 0.5

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

    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_binary640.6

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

    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. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left({\left(\frac{1}{{F}^{2} + 2}\right)}^{0.25} \cdot \frac{F}{\sin B}\right)} \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\]

    8. Simplified0.4

      \[\leadsto \color{blue}{\left({\left(\frac{1}{2 + F \cdot F}\right)}^{0.25} \cdot \frac{F}{\sin B}\right)} \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\]

    if 27842.903604652682 < 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. 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 simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.533745919767842 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 27842.90360465268:\\ \;\;\;\;\left({\left(\frac{1}{F \cdot F + 2}\right)}^{0.25} \cdot \frac{F}{\sin B}\right) \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right)\right) - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021166 
(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))))))