Average Error: 13.7 → 0.2
Time: 23.9s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \leq -46426.22023019213:\\ \;\;\;\;\left(\left(\frac{x}{{F}^{2} \cdot \sin B} + \frac{1}{{F}^{2} \cdot \sin B}\right) - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 32687143.65233844:\\ \;\;\;\;\frac{F}{\sin B} \cdot e^{-0.5 \cdot \log \left(x \cdot 2 + \left(2 + F \cdot F\right)\right)} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -46426.22023019213:\\
\;\;\;\;\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}\\

\mathbf{elif}\;F \leq 32687143.65233844:\\
\;\;\;\;\frac{F}{\sin B} \cdot e^{-0.5 \cdot \log \left(x \cdot 2 + \left(2 + F \cdot F\right)\right)} - \frac{x}{\tan 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 -46426.22023019213)
   (-
    (-
     (+ (/ x (* (pow F 2.0) (sin B))) (/ 1.0 (* (pow F 2.0) (sin B))))
     (/ 1.0 (sin B)))
    (/ x (tan B)))
   (if (<= F 32687143.65233844)
     (-
      (* (/ F (sin B)) (exp (* -0.5 (log (+ (* x 2.0) (+ 2.0 (* F F)))))))
      (/ x (tan 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 <= -46426.22023019213) {
		tmp = (((x / (pow(F, 2.0) * sin(B))) + (1.0 / (pow(F, 2.0) * sin(B)))) - (1.0 / sin(B))) - (x / tan(B));
	} else if (F <= 32687143.65233844) {
		tmp = ((F / sin(B)) * exp(-0.5 * log((x * 2.0) + (2.0 + (F * F))))) - (x / tan(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 < -46426.2202301921279

    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. Simplified25.2

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

    3. Taylor expanded around -inf 0.2

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

    if -46426.2202301921279 < F < 32687143.6523384415

    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 pow-to-exp_binary640.3

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

    5. Simplified0.3

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

    if 32687143.6523384415 < F

    1. Initial program 25.0

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

    2. Simplified24.9

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

    3. Taylor expanded around inf 0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -46426.22023019213:\\ \;\;\;\;\left(\left(\frac{x}{{F}^{2} \cdot \sin B} + \frac{1}{{F}^{2} \cdot \sin B}\right) - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 32687143.65233844:\\ \;\;\;\;\frac{F}{\sin B} \cdot e^{-0.5 \cdot \log \left(x \cdot 2 + \left(2 + F \cdot F\right)\right)} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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))))))