Average Error: 16.2 → 12.0
Time: 25.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{1 \cdot \tan \left(\pi \cdot \ell\right)}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{1 \cdot \tan \left(\pi \cdot \ell\right)}}}{F}
double f(double F, double l) {
        double r24116 = atan2(1.0, 0.0);
        double r24117 = l;
        double r24118 = r24116 * r24117;
        double r24119 = 1.0;
        double r24120 = F;
        double r24121 = r24120 * r24120;
        double r24122 = r24119 / r24121;
        double r24123 = tan(r24118);
        double r24124 = r24122 * r24123;
        double r24125 = r24118 - r24124;
        return r24125;
}
↓
double f(double F, double l) {
        double r24126 = atan2(1.0, 0.0);
        double r24127 = l;
        double r24128 = r24126 * r24127;
        double r24129 = 1.0;
        double r24130 = F;
        double r24131 = 1.0;
        double r24132 = tan(r24128);
        double r24133 = r24131 * r24132;
        double r24134 = r24130 / r24133;
        double r24135 = r24129 / r24134;
        double r24136 = r24135 / r24130;
        double r24137 = r24128 - r24136;
        return r24137;
}

Error

Bits error versus F

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.2

    \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity16.2

    \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot 1}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

  4. Applied times-frac16.2

    \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F} \cdot \frac{1}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

  5. Applied associate-*l*12.0

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
  6. Using strategy rm

  7. Applied associate-*l/12.0

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity12.0

    \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{1 \cdot F}} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\]

  10. Applied add-sqr-sqrt12.0

    \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot F} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\]

  11. Applied times-frac12.0

    \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{F}\right)} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\]

  12. Applied associate-*l*12.0

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{F} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)}\]

  13. Simplified12.0

    \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{1} \cdot \color{blue}{\frac{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
  14. Using strategy rm

  15. Applied clear-num12.0

    \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{1} \cdot \frac{\color{blue}{\frac{1}{\frac{F}{1 \cdot \tan \left(\pi \cdot \ell\right)}}}}{F}\]

  16. Final simplification12.0

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{1 \cdot \tan \left(\pi \cdot \ell\right)}}}{F}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))