Average Error: 13.8 → 10.9
Time: 32.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{\cos B \cdot x}{\sin B}\]
\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)}
↓
\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{\cos B \cdot x}{\sin B}
double f(double F, double B, double x) {
        double r672550 = x;
        double r672551 = 1.0;
        double r672552 = B;
        double r672553 = tan(r672552);
        double r672554 = r672551 / r672553;
        double r672555 = r672550 * r672554;
        double r672556 = -r672555;
        double r672557 = F;
        double r672558 = sin(r672552);
        double r672559 = r672557 / r672558;
        double r672560 = r672557 * r672557;
        double r672561 = 2.0;
        double r672562 = r672560 + r672561;
        double r672563 = r672561 * r672550;
        double r672564 = r672562 + r672563;
        double r672565 = r672551 / r672561;
        double r672566 = -r672565;
        double r672567 = pow(r672564, r672566);
        double r672568 = r672559 * r672567;
        double r672569 = r672556 + r672568;
        return r672569;
}
↓
double f(double F, double B, double x) {
        double r672570 = 1.0;
        double r672571 = B;
        double r672572 = sin(r672571);
        double r672573 = r672570 / r672572;
        double r672574 = x;
        double r672575 = 2.0;
        double r672576 = F;
        double r672577 = fma(r672576, r672576, r672575);
        double r672578 = fma(r672574, r672575, r672577);
        double r672579 = -0.5;
        double r672580 = pow(r672578, r672579);
        double r672581 = r672580 * r672576;
        double r672582 = r672573 * r672581;
        double r672583 = cos(r672571);
        double r672584 = r672583 * r672574;
        double r672585 = r672584 / r672572;
        double r672586 = r672582 - r672585;
        return r672586;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.8

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

  2. Simplified13.7

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

  4. Applied div-inv13.7

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

  5. Applied associate-*r*10.8

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

  6. Taylor expanded around inf 10.9

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

  7. Final simplification10.9

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))