Average Error: 13.7 → 0.2
Time: 11.9s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.3661477811092286 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 37847795.092115104:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sin B \cdot {F}^{2}\\ \left(\frac{1}{\sin B} - \left(\frac{x}{t_1} + \frac{1}{t_1}\right)\right) - t_0 \end{array}\\ \end{array} \]
\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}\\
\mathbf{if}\;F \leq -3.3661477811092286 \cdot 10^{+22}:\\
\;\;\;\;\frac{-1}{\sin B} - t_0\\

\mathbf{elif}\;F \leq 37847795.092115104:\\
\;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - t_0\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \sin B \cdot {F}^{2}\\
\left(\frac{1}{\sin B} - \left(\frac{x}{t_1} + \frac{1}{t_1}\right)\right) - t_0
\end{array}\\


\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))))
   (if (<= F -3.3661477811092286e+22)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 37847795.092115104)
       (- (* (/ 1.0 (/ (sin B) F)) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) t_0)
       (let* ((t_1 (* (sin B) (pow F 2.0))))
         (- (- (/ 1.0 (sin B)) (+ (/ x t_1) (/ 1.0 t_1))) 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 tmp;
	if (F <= -3.3661477811092286e+22) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 37847795.092115104) {
		tmp = ((1.0 / (sin(B) / F)) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - t_0;
	} else {
		double t_1 = sin(B) * pow(F, 2.0);
		tmp = ((1.0 / sin(B)) - ((x / t_1) + (1.0 / t_1))) - t_0;
	}
	return tmp;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -3.36614778110922856e22

    1. Initial program 26.5

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

    2. Simplified26.5

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

    3. Taylor expanded in F around -inf 0.2

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

    if -3.36614778110922856e22 < F < 37847795.092115104

    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

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

    3. Applied clear-num_binary640.3

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

    if 37847795.092115104 < F

    1. Initial program 25.6

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

    2. Simplified25.5

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

    3. Taylor expanded in F around inf 0.1

      \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot {F}^{2}} + \frac{1}{\sin B \cdot {F}^{2}}\right)\right)} - \frac{x}{\tan B} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3661477811092286 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 37847795.092115104:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\sin B \cdot {F}^{2}} + \frac{1}{\sin B \cdot {F}^{2}}\right)\right) - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))))