Average Error: 13.3 → 10.4
Time: 51.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{2}} \cdot F}{\sin B} - \cos B \cdot \frac{x}{\sin B}\]
\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{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{2}} \cdot F}{\sin B} - \cos B \cdot \frac{x}{\sin B}
double f(double F, double B, double x) {
        double r2743799 = x;
        double r2743800 = 1.0;
        double r2743801 = B;
        double r2743802 = tan(r2743801);
        double r2743803 = r2743800 / r2743802;
        double r2743804 = r2743799 * r2743803;
        double r2743805 = -r2743804;
        double r2743806 = F;
        double r2743807 = sin(r2743801);
        double r2743808 = r2743806 / r2743807;
        double r2743809 = r2743806 * r2743806;
        double r2743810 = 2.0;
        double r2743811 = r2743809 + r2743810;
        double r2743812 = r2743810 * r2743799;
        double r2743813 = r2743811 + r2743812;
        double r2743814 = r2743800 / r2743810;
        double r2743815 = -r2743814;
        double r2743816 = pow(r2743813, r2743815);
        double r2743817 = r2743808 * r2743816;
        double r2743818 = r2743805 + r2743817;
        return r2743818;
}
↓
double f(double F, double B, double x) {
        double r2743819 = 2.0;
        double r2743820 = x;
        double r2743821 = r2743819 * r2743820;
        double r2743822 = F;
        double r2743823 = r2743822 * r2743822;
        double r2743824 = r2743821 + r2743823;
        double r2743825 = r2743824 + r2743819;
        double r2743826 = -0.5;
        double r2743827 = pow(r2743825, r2743826);
        double r2743828 = r2743827 * r2743822;
        double r2743829 = B;
        double r2743830 = sin(r2743829);
        double r2743831 = r2743828 / r2743830;
        double r2743832 = cos(r2743829);
        double r2743833 = r2743820 / r2743830;
        double r2743834 = r2743832 * r2743833;
        double r2743835 = r2743831 - r2743834;
        return r2743835;
}

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. Initial program 13.3

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

  2. Simplified10.3

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

  4. Applied tan-quot10.4

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

  5. Applied associate-/r/10.4

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

  6. Final simplification10.4

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

Reproduce

herbie shell --seed 2019139 
(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))))))