Average Error: 13.2 → 0.6
Time: 12.1s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -4.5733962104717876 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.870789093939713 \cdot 10^{-09}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -4.5733962104717876 \cdot 10^{+49}:\\
\;\;\;\;\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 6.870789093939713 \cdot 10^{-09}:\\
\;\;\;\;\frac{F}{\sin B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x \cdot \cos B}{\sin 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 -4.5733962104717876e+49)
   (-
    (+
     (+ (/ x (* (* F F) (sin B))) (/ 1.0 (* (* F F) (sin B))))
     (/ -1.0 (sin B)))
    (/ x (tan B)))
   (if (<= F 6.870789093939713e-09)
     (-
      (/ F (* (sin B) (sqrt (+ (* x 2.0) (+ (* F F) 2.0)))))
      (/ (* x (cos B)) (sin 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 <= -4.5733962104717876e+49) {
		tmp = (((x / ((F * F) * sin(B))) + (1.0 / ((F * F) * sin(B)))) + (-1.0 / sin(B))) - (x / tan(B));
	} else if (F <= 6.870789093939713e-09) {
		tmp = (F / (sin(B) * sqrt((x * 2.0) + ((F * F) + 2.0)))) - ((x * cos(B)) / sin(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 < -4.57339621047178755e49

    1. Initial program 28.0

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

    2. Simplified27.9

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -4.57339621047178755e49 < F < 6.87078909393971308e-9

    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. Using strategy rm

    3. Applied neg-sub0_binary640.5

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(0 - \frac{1}{2}\right)}}\]

    4. Applied pow-sub_binary640.6

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{0}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}\]

    5. Applied frac-times_binary640.4

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{0}}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}\]

    6. Simplified0.4

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

    7. Simplified0.4

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

    8. Taylor expanded around 0 0.3

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

    if 6.87078909393971308e-9 < F

    1. Initial program 23.7

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

    2. Simplified23.7

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

    3. Taylor expanded around inf 1.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.5733962104717876 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.870789093939713 \cdot 10^{-09}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))))))