Average Error: 13.9 → 0.2
Time: 42.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -5105757622.280994415283203125:\\ \;\;\;\;\left(\frac{1}{\left(F \cdot F\right) \cdot \sin B} - \frac{1}{\sin B}\right) - \frac{1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \le 15322.32932603809967986308038234710693359:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{{\left(\left(2 + x \cdot 2\right) + F \cdot F\right)}^{\left(\frac{1}{2}\right)}}}} - \frac{1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) - \frac{1 \cdot x}{\tan B}\\ \end{array}\]
\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 \le -5105757622.280994415283203125:\\
\;\;\;\;\left(\frac{1}{\left(F \cdot F\right) \cdot \sin B} - \frac{1}{\sin B}\right) - \frac{1 \cdot x}{\tan B}\\

\mathbf{elif}\;F \le 15322.32932603809967986308038234710693359:\\
\;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{{\left(\left(2 + x \cdot 2\right) + F \cdot F\right)}^{\left(\frac{1}{2}\right)}}}} - \frac{1 \cdot x}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r103463 = x;
        double r103464 = 1.0;
        double r103465 = B;
        double r103466 = tan(r103465);
        double r103467 = r103464 / r103466;
        double r103468 = r103463 * r103467;
        double r103469 = -r103468;
        double r103470 = F;
        double r103471 = sin(r103465);
        double r103472 = r103470 / r103471;
        double r103473 = r103470 * r103470;
        double r103474 = 2.0;
        double r103475 = r103473 + r103474;
        double r103476 = r103474 * r103463;
        double r103477 = r103475 + r103476;
        double r103478 = r103464 / r103474;
        double r103479 = -r103478;
        double r103480 = pow(r103477, r103479);
        double r103481 = r103472 * r103480;
        double r103482 = r103469 + r103481;
        return r103482;
}
↓
double f(double F, double B, double x) {
        double r103483 = F;
        double r103484 = -5105757622.280994;
        bool r103485 = r103483 <= r103484;
        double r103486 = 1.0;
        double r103487 = r103483 * r103483;
        double r103488 = B;
        double r103489 = sin(r103488);
        double r103490 = r103487 * r103489;
        double r103491 = r103486 / r103490;
        double r103492 = 1.0;
        double r103493 = r103492 / r103489;
        double r103494 = r103491 - r103493;
        double r103495 = x;
        double r103496 = r103486 * r103495;
        double r103497 = tan(r103488);
        double r103498 = r103496 / r103497;
        double r103499 = r103494 - r103498;
        double r103500 = 15322.3293260381;
        bool r103501 = r103483 <= r103500;
        double r103502 = 2.0;
        double r103503 = r103495 * r103502;
        double r103504 = r103502 + r103503;
        double r103505 = r103504 + r103487;
        double r103506 = r103486 / r103502;
        double r103507 = pow(r103505, r103506);
        double r103508 = r103492 / r103507;
        double r103509 = r103489 / r103508;
        double r103510 = r103483 / r103509;
        double r103511 = r103510 - r103498;
        double r103512 = r103493 - r103491;
        double r103513 = r103512 - r103498;
        double r103514 = r103501 ? r103511 : r103513;
        double r103515 = r103485 ? r103499 : r103514;
        return r103515;
}

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 < -5105757622.280994

    1. Initial program 25.4

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

    2. Simplified19.4

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

    4. Applied pow-neg19.4

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

    5. Simplified19.4

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

    6. Taylor expanded around -inf 0.2

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

    7. Simplified0.2

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

    if -5105757622.280994 < F < 15322.3293260381

    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

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

    4. Applied pow-neg0.3

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

    5. Simplified0.3

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

    if 15322.3293260381 < F

    1. Initial program 25.8

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

    2. Simplified20.0

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

    4. Applied pow-neg20.0

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

    5. Simplified20.0

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

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -5105757622.280994415283203125:\\ \;\;\;\;\left(\frac{1}{\left(F \cdot F\right) \cdot \sin B} - \frac{1}{\sin B}\right) - \frac{1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \le 15322.32932603809967986308038234710693359:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{{\left(\left(2 + x \cdot 2\right) + F \cdot F\right)}^{\left(\frac{1}{2}\right)}}}} - \frac{1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) - \frac{1 \cdot x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))