VandenBroeck and Keller, Equation (6)

Average Error: 16.9 → 12.9

Time: 28.2s

Precision: 64

• could not determine a ground truth for program body

  1. F = 9.664659838372436e+286
  2. l = -2497197479928642.5

Math

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

↓

\[\pi \cdot \ell - \left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \frac{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)\]

C

double f(double F, double l) {
        double r30053 = atan2(1.0, 0.0);
        double r30054 = l;
        double r30055 = r30053 * r30054;
        double r30056 = 1.0;
        double r30057 = F;
        double r30058 = r30057 * r30057;
        double r30059 = r30056 / r30058;
        double r30060 = tan(r30055);
        double r30061 = r30059 * r30060;
        double r30062 = r30055 - r30061;
        return r30062;
}

↓

double f(double F, double l) {
        double r30063 = atan2(1.0, 0.0);
        double r30064 = l;
        double r30065 = r30063 * r30064;
        double r30066 = 1.0;
        double r30067 = cbrt(r30066);
        double r30068 = r30067 * r30067;
        double r30069 = F;
        double r30070 = r30068 / r30069;
        double r30071 = cbrt(r30070);
        double r30072 = r30071 * r30071;
        double r30073 = tan(r30065);
        double r30074 = r30067 * r30073;
        double r30075 = r30074 / r30069;
        double r30076 = r30071 * r30075;
        double r30077 = r30072 * r30076;
        double r30078 = r30065 - r30077;
        return r30078;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Initial program 16.9

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

3. Applied add-cube-cbrt 16.9

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

4. Applied times-frac 16.9

   \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

5. Applied associate-*l* 12.7

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
6. Using strategy rm

7. Applied *-un-lft-identity 12.7

   \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\color{blue}{\left(1 \cdot \frac{\sqrt[3]{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\right)\]

8. Applied associate-*l* 12.7

   \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \color{blue}{\left(1 \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)}\]

9. Simplified 12.7

   \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(1 \cdot \color{blue}{\frac{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}}\right)\]
10. Using strategy rm

11. Applied add-cube-cbrt 12.9

    \[\leadsto \pi \cdot \ell - \color{blue}{\left(\left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}}\right)} \cdot \left(1 \cdot \frac{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)\]

12. Applied associate-*l* 12.9

    \[\leadsto \pi \cdot \ell - \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \left(1 \cdot \frac{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)\right)}\]

13. Simplified 12.9

    \[\leadsto \pi \cdot \ell - \left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}}\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \frac{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)}\]

14. Final simplification 12.9

    \[\leadsto \pi \cdot \ell - \left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}} \cdot \frac{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))