Average Error: 13.9 → 0.2
Time: 10.0s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -1.0378427485936736 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 51281.2123343461:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -1.0378427485936736 \cdot 10^{+18}:\\
\;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 51281.2123343461:\\
\;\;\;\;\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\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 -1.0378427485936736e+18)
   (- (/ (- (/ 1.0 (* F F)) 1.0) (sin B)) (/ x (tan B)))
   (if (<= F 51281.2123343461)
     (-
      (/ F (/ (sin B) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)))
      (/ x (tan B)))
     (- (/ (- 1.0 (/ 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 <= -1.0378427485936736e+18) {
		tmp = (((1.0 / (F * F)) - 1.0) / sin(B)) - (x / tan(B));
	} else if (F <= 51281.2123343461) {
		tmp = (F / (sin(B) / pow((((F * F) + 2.0) + (x * 2.0)), -0.5))) - (x / tan(B));
	} else {
		tmp = ((1.0 - (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 < -1037842748593673600

    1. Initial program 25.5

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

    2. Simplified25.5

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

    4. Applied associate-*l/_binary6419.3

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

    5. Simplified19.3

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

    6. Taylor expanded around -inf 0.2

      \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} - 1}}{\sin B} - \frac{x}{\tan B}\]

    7. Simplified0.2

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

    if -1037842748593673600 < F < 51281.212334346099

    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. Simplified0.4

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

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

    7. Applied associate-/l*_binary640.3

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

    if 51281.212334346099 < F

    1. Initial program 26.2

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

    2. Simplified26.1

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

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

    6. Taylor expanded around inf 0.2

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{{F}^{2}}}}{\sin B} - \frac{x}{\tan B}\]

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.0378427485936736 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 51281.2123343461:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))))))