Average Error: 14.0 → 0.2
Time: 12.4s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \left(F \cdot F\right) \cdot \sin B\\ t_2 := \frac{x}{t_1} + \frac{1}{t_1}\\ \mathbf{if}\;F \leq -1525898.2560680246:\\ \;\;\;\;\left(t_2 + \frac{-1}{\sin B}\right) - t_0\\ \mathbf{elif}\;F \leq 97379942.01309888:\\ \;\;\;\;\left(F \cdot {\left(2 + \left(F \cdot F + x \cdot 2\right)\right)}^{-0.25}\right) \cdot \frac{\sqrt{\sqrt{\frac{1}{F \cdot F + 2}}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - t_2\right) - t_0\\ \end{array} \]
\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}
t_0 := \frac{x}{\tan B}\\
t_1 := \left(F \cdot F\right) \cdot \sin B\\
t_2 := \frac{x}{t_1} + \frac{1}{t_1}\\
\mathbf{if}\;F \leq -1525898.2560680246:\\
\;\;\;\;\left(t_2 + \frac{-1}{\sin B}\right) - t_0\\

\mathbf{elif}\;F \leq 97379942.01309888:\\
\;\;\;\;\left(F \cdot {\left(2 + \left(F \cdot F + x \cdot 2\right)\right)}^{-0.25}\right) \cdot \frac{\sqrt{\sqrt{\frac{1}{F \cdot F + 2}}}}{\sin B} - t_0\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{\sin B} - t_2\right) - t_0\\


\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
 (let* ((t_0 (/ x (tan B)))
        (t_1 (* (* F F) (sin B)))
        (t_2 (+ (/ x t_1) (/ 1.0 t_1))))
   (if (<= F -1525898.2560680246)
     (- (+ t_2 (/ -1.0 (sin B))) t_0)
     (if (<= F 97379942.01309888)
       (-
        (*
         (* F (pow (+ 2.0 (+ (* F F) (* x 2.0))) -0.25))
         (/ (sqrt (sqrt (/ 1.0 (+ (* F F) 2.0)))) (sin B)))
        t_0)
       (- (- (/ 1.0 (sin B)) t_2) t_0)))))
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 t_0 = x / tan(B);
	double t_1 = (F * F) * sin(B);
	double t_2 = (x / t_1) + (1.0 / t_1);
	double tmp;
	if (F <= -1525898.2560680246) {
		tmp = (t_2 + (-1.0 / sin(B))) - t_0;
	} else if (F <= 97379942.01309888) {
		tmp = ((F * pow((2.0 + ((F * F) + (x * 2.0))), -0.25)) * (sqrt(sqrt(1.0 / ((F * F) + 2.0))) / sin(B))) - t_0;
	} else {
		tmp = ((1.0 / sin(B)) - t_2) - t_0;
	}
	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 < -1525898.25606802455

    1. Initial program 25.3

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

    2. Simplified25.3

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

    3. Taylor expanded around -inf 0.2

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

    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 -1525898.25606802455 < F < 97379942.0130988806

    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_binary640.3

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

    5. Applied associate-*l*_binary640.3

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

    6. Simplified0.3

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

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

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

    9. Applied add-sqr-sqrt_binary640.6

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

    10. Applied times-frac_binary640.5

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

    11. Applied associate-*r*_binary640.5

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

    12. Simplified0.3

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

    13. Taylor expanded around 0 0.3

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

    14. Simplified0.3

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

    if 97379942.0130988806 < 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. Simplified25.9

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1525898.2560680246:\\ \;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 97379942.01309888:\\ \;\;\;\;\left(F \cdot {\left(2 + \left(F \cdot F + x \cdot 2\right)\right)}^{-0.25}\right) \cdot \frac{\sqrt{\sqrt{\frac{1}{F \cdot F + 2}}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right)\right) - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2021197 
(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))))))