Average Error: 13.5 → 10.8
Time: 1.6m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{1}{\sin B} \cdot \left({\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}\]
\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(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r2419947 = x;
        double r2419948 = 1.0;
        double r2419949 = B;
        double r2419950 = tan(r2419949);
        double r2419951 = r2419948 / r2419950;
        double r2419952 = r2419947 * r2419951;
        double r2419953 = -r2419952;
        double r2419954 = F;
        double r2419955 = sin(r2419949);
        double r2419956 = r2419954 / r2419955;
        double r2419957 = r2419954 * r2419954;
        double r2419958 = 2.0;
        double r2419959 = r2419957 + r2419958;
        double r2419960 = r2419958 * r2419947;
        double r2419961 = r2419959 + r2419960;
        double r2419962 = r2419948 / r2419958;
        double r2419963 = -r2419962;
        double r2419964 = pow(r2419961, r2419963);
        double r2419965 = r2419956 * r2419964;
        double r2419966 = r2419953 + r2419965;
        return r2419966;
}
↓
double f(double F, double B, double x) {
        double r2419967 = 1.0;
        double r2419968 = B;
        double r2419969 = sin(r2419968);
        double r2419970 = r2419967 / r2419969;
        double r2419971 = 2.0;
        double r2419972 = x;
        double r2419973 = r2419971 * r2419972;
        double r2419974 = F;
        double r2419975 = r2419974 * r2419974;
        double r2419976 = r2419975 + r2419971;
        double r2419977 = r2419973 + r2419976;
        double r2419978 = -0.5;
        double r2419979 = pow(r2419977, r2419978);
        double r2419980 = r2419979 * r2419974;
        double r2419981 = r2419970 * r2419980;
        double r2419982 = tan(r2419968);
        double r2419983 = r2419972 / r2419982;
        double r2419984 = r2419981 - r2419983;
        return r2419984;
}

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 13.5

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

  2. Simplified13.1

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

  4. Applied div-inv13.1

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

  5. Applied *-un-lft-identity13.1

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

  6. Applied *-un-lft-identity13.1

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

  7. Applied distribute-lft-out13.1

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

  8. Applied unpow-prod-down13.1

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

  9. Applied times-frac10.8

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

  10. Simplified10.8

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

  11. Simplified10.8

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

  12. Final simplification10.8

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

Reproduce

herbie shell --seed 2019142 
(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))))))