Average Error: 14.2 → 0.2
Time: 14.4s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -64003883249.95031:\\ \;\;\;\;\frac{1}{-\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9689563051736543 \cdot 10^{+26}:\\ \;\;\;\;\left(F \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\sin B}{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -64003883249.95031:\\
\;\;\;\;\frac{1}{-\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 1.9689563051736543 \cdot 10^{+26}:\\
\;\;\;\;\left(F \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\sin B}{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \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 -64003883249.95031)
   (- (/ 1.0 (- (sin B))) (/ x (tan B)))
   (if (<= F 1.9689563051736543e+26)
     (-
      (*
       (* F (* (cbrt 1.0) (cbrt 1.0)))
       (/ (cbrt 1.0) (/ (sin B) (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5))))
      (/ 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 <= -64003883249.95031) {
		tmp = (1.0 / -sin(B)) - (x / tan(B));
	} else if (F <= 1.9689563051736543e+26) {
		tmp = ((F * (cbrt(1.0) * cbrt(1.0))) * (cbrt(1.0) / (sin(B) / pow(((x * 2.0) + (2.0 + (F * F))), -0.5)))) - (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 < -64003883249.9503098

    1. Initial program 25.8

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

    2. Simplified25.8

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

    4. Applied associate-*l/_binary6419.8

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

    5. Simplified19.8

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

    7. Applied clear-num_binary6419.9

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

    8. Simplified19.9

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

    9. Taylor expanded around -inf 0.2

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

    10. Simplified0.2

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

    if -64003883249.9503098 < F < 1.9689563051736543e26

    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 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

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

    7. Applied clear-num_binary640.3

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

    8. Simplified0.3

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

    10. Applied *-un-lft-identity_binary640.3

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

    11. Applied times-frac_binary640.3

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

    12. Applied add-cube-cbrt_binary640.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{F} \cdot \frac{\sin B}{{\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\]

    13. Applied times-frac_binary640.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{F}} \cdot \frac{\sqrt[3]{1}}{\frac{\sin B}{{\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{-0.5}}}} - \frac{x}{\tan B}\]

    14. Simplified0.3

      \[\leadsto \color{blue}{\left(F \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)} \cdot \frac{\sqrt[3]{1}}{\frac{\sin B}{{\left(F \cdot F + \left(2 + 2 \cdot x\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\]

    15. Simplified0.3

      \[\leadsto \left(F \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sin B}{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}}} - \frac{x}{\tan B}\]

    if 1.9689563051736543e26 < F

    1. Initial program 27.8

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

    2. Simplified27.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -64003883249.95031:\\ \;\;\;\;\frac{1}{-\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9689563051736543 \cdot 10^{+26}:\\ \;\;\;\;\left(F \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{\sin B}{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021113 
(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))))))