Average Error: 13.6 → 0.3
Time: 12.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2.393293982823570639154866165724748883128 \cdot 10^{51}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{\frac{1}{F}}{F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 16347121045823049728:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1 - \frac{\frac{1}{F}}{F}}{\sin B}\\ \end{array}\]
\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 \le -2.393293982823570639154866165724748883128 \cdot 10^{51}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{\frac{1}{F}}{F} - 1}{\sin B}\\

\mathbf{elif}\;F \le 16347121045823049728:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \frac{1}{\sin B}\\

\mathbf{else}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1 - \frac{\frac{1}{F}}{F}}{\sin B}\\

\end{array}
double f(double F, double B, double x) {
        double r40551 = x;
        double r40552 = 1.0;
        double r40553 = B;
        double r40554 = tan(r40553);
        double r40555 = r40552 / r40554;
        double r40556 = r40551 * r40555;
        double r40557 = -r40556;
        double r40558 = F;
        double r40559 = sin(r40553);
        double r40560 = r40558 / r40559;
        double r40561 = r40558 * r40558;
        double r40562 = 2.0;
        double r40563 = r40561 + r40562;
        double r40564 = r40562 * r40551;
        double r40565 = r40563 + r40564;
        double r40566 = r40552 / r40562;
        double r40567 = -r40566;
        double r40568 = pow(r40565, r40567);
        double r40569 = r40560 * r40568;
        double r40570 = r40557 + r40569;
        return r40570;
}
↓
double f(double F, double B, double x) {
        double r40571 = F;
        double r40572 = -2.3932939828235706e+51;
        bool r40573 = r40571 <= r40572;
        double r40574 = x;
        double r40575 = 1.0;
        double r40576 = r40574 * r40575;
        double r40577 = B;
        double r40578 = tan(r40577);
        double r40579 = r40576 / r40578;
        double r40580 = -r40579;
        double r40581 = r40575 / r40571;
        double r40582 = r40581 / r40571;
        double r40583 = 1.0;
        double r40584 = r40582 - r40583;
        double r40585 = sin(r40577);
        double r40586 = r40584 / r40585;
        double r40587 = r40580 + r40586;
        double r40588 = 1.634712104582305e+19;
        bool r40589 = r40571 <= r40588;
        double r40590 = cos(r40577);
        double r40591 = r40574 * r40590;
        double r40592 = r40591 / r40585;
        double r40593 = r40575 * r40592;
        double r40594 = -r40593;
        double r40595 = r40571 * r40571;
        double r40596 = 2.0;
        double r40597 = r40595 + r40596;
        double r40598 = r40596 * r40574;
        double r40599 = r40597 + r40598;
        double r40600 = r40575 / r40596;
        double r40601 = -r40600;
        double r40602 = pow(r40599, r40601);
        double r40603 = r40571 * r40602;
        double r40604 = r40583 / r40585;
        double r40605 = r40603 * r40604;
        double r40606 = r40594 + r40605;
        double r40607 = r40583 - r40582;
        double r40608 = r40607 / r40585;
        double r40609 = r40580 + r40608;
        double r40610 = r40589 ? r40606 : r40609;
        double r40611 = r40573 ? r40587 : r40610;
        return r40611;
}

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 < -2.3932939828235706e+51

    1. Initial program 28.1

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

    3. Applied associate-*l/21.4

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

    5. Applied associate-*r/21.4

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

    6. Taylor expanded around -inf 0.2

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

    7. Simplified0.2

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

    if -2.3932939828235706e+51 < F < 1.634712104582305e+19

    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 associate-*l/0.4

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

    5. Applied associate-*r/0.3

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

    7. Applied div-inv0.3

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

    8. Taylor expanded around inf 0.4

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

    if 1.634712104582305e+19 < F

    1. Initial program 26.6

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

    3. Applied associate-*l/19.9

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

    5. Applied associate-*r/19.8

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

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -2.393293982823570639154866165724748883128 \cdot 10^{51}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{\frac{1}{F}}{F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 16347121045823049728:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1 - \frac{\frac{1}{F}}{F}}{\sin B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))