Average Error: 13.5 → 13.5
Time: 35.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\frac{1}{2}\right)}, F \cdot \frac{1}{\sin B}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\]
\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)}
↓
\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\frac{1}{2}\right)}, F \cdot \frac{1}{\sin B}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)
double f(double F, double B, double x) {
        double r51608 = x;
        double r51609 = 1.0;
        double r51610 = B;
        double r51611 = tan(r51610);
        double r51612 = r51609 / r51611;
        double r51613 = r51608 * r51612;
        double r51614 = -r51613;
        double r51615 = F;
        double r51616 = sin(r51610);
        double r51617 = r51615 / r51616;
        double r51618 = r51615 * r51615;
        double r51619 = 2.0;
        double r51620 = r51618 + r51619;
        double r51621 = r51619 * r51608;
        double r51622 = r51620 + r51621;
        double r51623 = r51609 / r51619;
        double r51624 = -r51623;
        double r51625 = pow(r51622, r51624);
        double r51626 = r51617 * r51625;
        double r51627 = r51614 + r51626;
        return r51627;
}
↓
double f(double F, double B, double x) {
        double r51628 = 2.0;
        double r51629 = x;
        double r51630 = F;
        double r51631 = fma(r51630, r51630, r51628);
        double r51632 = fma(r51628, r51629, r51631);
        double r51633 = 1.0;
        double r51634 = r51633 / r51628;
        double r51635 = -r51634;
        double r51636 = pow(r51632, r51635);
        double r51637 = 1.0;
        double r51638 = B;
        double r51639 = sin(r51638);
        double r51640 = r51637 / r51639;
        double r51641 = r51630 * r51640;
        double r51642 = cos(r51638);
        double r51643 = r51629 * r51642;
        double r51644 = r51643 / r51639;
        double r51645 = r51633 * r51644;
        double r51646 = -r51645;
        double r51647 = fma(r51636, r51641, r51646);
        return r51647;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.5

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

  2. Simplified13.5

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

  4. Applied associate-*r/13.4

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

  6. Applied div-inv13.4

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

  7. Taylor expanded around inf 13.5

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

  8. Final simplification13.5

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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))))))