Average Error: 13.5 → 10.7
Time: 47.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\sin B} \cdot {\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\frac{1}{F}} - \frac{x}{\tan B}\]
\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{\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\sin B} \cdot {\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\frac{1}{F}} - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r1514945 = x;
        double r1514946 = 1.0;
        double r1514947 = B;
        double r1514948 = tan(r1514947);
        double r1514949 = r1514946 / r1514948;
        double r1514950 = r1514945 * r1514949;
        double r1514951 = -r1514950;
        double r1514952 = F;
        double r1514953 = sin(r1514947);
        double r1514954 = r1514952 / r1514953;
        double r1514955 = r1514952 * r1514952;
        double r1514956 = 2.0;
        double r1514957 = r1514955 + r1514956;
        double r1514958 = r1514956 * r1514945;
        double r1514959 = r1514957 + r1514958;
        double r1514960 = r1514946 / r1514956;
        double r1514961 = -r1514960;
        double r1514962 = pow(r1514959, r1514961);
        double r1514963 = r1514954 * r1514962;
        double r1514964 = r1514951 + r1514963;
        return r1514964;
}
↓
double f(double F, double B, double x) {
        double r1514965 = 2.0;
        double r1514966 = x;
        double r1514967 = r1514965 * r1514966;
        double r1514968 = F;
        double r1514969 = r1514968 * r1514968;
        double r1514970 = r1514967 + r1514969;
        double r1514971 = r1514970 + r1514965;
        double r1514972 = -0.25;
        double r1514973 = pow(r1514971, r1514972);
        double r1514974 = B;
        double r1514975 = sin(r1514974);
        double r1514976 = r1514973 / r1514975;
        double r1514977 = r1514976 * r1514973;
        double r1514978 = 1.0;
        double r1514979 = r1514978 / r1514968;
        double r1514980 = r1514977 / r1514979;
        double r1514981 = tan(r1514974);
        double r1514982 = r1514966 / r1514981;
        double r1514983 = r1514980 - r1514982;
        return r1514983;
}

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. Simplified12.9

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

  4. Applied div-inv12.9

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

  5. Applied associate-/r*10.6

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

  7. Applied *-un-lft-identity10.6

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

  8. Applied sqr-pow10.7

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

  9. Applied times-frac10.7

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

  10. Final simplification10.7

    \[\leadsto \frac{\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\sin B} \cdot {\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\frac{1}{F}} - \frac{x}{\tan B}\]

Reproduce

herbie shell --seed 2019146 
(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))))))