Average Error: 13.6 → 0.2
Time: 13.1s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.634419849664236 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.586080114579332 \cdot 10^{+92}:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - t_0\\ \end{array} \]
\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}\\
\mathbf{if}\;F \leq -1.634419849664236 \cdot 10^{+126}:\\
\;\;\;\;\frac{-1}{\sin B} - t_0\\

\mathbf{elif}\;F \leq 6.586080114579332 \cdot 10^{+92}:\\
\;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\

\mathbf{else}:\\
\;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - 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))))
   (if (<= F -1.634419849664236e+126)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 6.586080114579332e+92)
       (- (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)) t_0)
       (- (* F (/ 1.0 (* F (sin B)))) 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 <= -1.634419849664236e+126) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 6.586080114579332e+92) {
		tmp = ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B)) - t_0;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - 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 < -1.63441984966423587e126

    1. Initial program 37.7

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

    2. Simplified37.6

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

    3. Taylor expanded in F around -inf 0.2

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

    if -1.63441984966423587e126 < F < 6.5860801145793315e92

    1. Initial program 1.6

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

    2. Simplified1.5

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

    3. Applied div-inv_binary641.5

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

    4. Applied associate-*l*_binary640.3

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

    5. Simplified0.3

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

    6. Applied add-sqr-sqrt_binary640.3

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

    7. Applied unpow-prod-down_binary640.3

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

    8. Applied associate-*r/_binary640.3

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

    9. Simplified0.3

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

    10. Applied pow1_binary640.3

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

    11. Applied sqrt-pow1_binary640.3

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

    12. Applied pow-flip_binary640.3

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

    if 6.5860801145793315e92 < F

    1. Initial program 33.0

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

    2. Simplified32.9

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

    3. Applied div-inv_binary6432.9

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

    4. Applied associate-*l*_binary6426.6

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

    5. Simplified26.6

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

    6. Taylor expanded in F around inf 0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.634419849664236 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.586080114579332 \cdot 10^{+92}:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022094 
(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))))))