Average Error: 0.4 → 0.4
Time: 24.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\left(\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{1}{\left(\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r203801 = 1.0;
        double r203802 = 5.0;
        double r203803 = v;
        double r203804 = r203803 * r203803;
        double r203805 = r203802 * r203804;
        double r203806 = r203801 - r203805;
        double r203807 = atan2(1.0, 0.0);
        double r203808 = t;
        double r203809 = r203807 * r203808;
        double r203810 = 2.0;
        double r203811 = 3.0;
        double r203812 = r203811 * r203804;
        double r203813 = r203801 - r203812;
        double r203814 = r203810 * r203813;
        double r203815 = sqrt(r203814);
        double r203816 = r203809 * r203815;
        double r203817 = r203801 - r203804;
        double r203818 = r203816 * r203817;
        double r203819 = r203806 / r203818;
        return r203819;
}
↓
double f(double v, double t) {
        double r203820 = 1.0;
        double r203821 = t;
        double r203822 = 2.0;
        double r203823 = sqrt(r203822);
        double r203824 = atan2(1.0, 0.0);
        double r203825 = r203823 * r203824;
        double r203826 = r203821 * r203825;
        double r203827 = 3.0;
        double r203828 = v;
        double r203829 = r203828 * r203828;
        double r203830 = r203827 * r203829;
        double r203831 = r203820 - r203830;
        double r203832 = sqrt(r203831);
        double r203833 = r203826 * r203832;
        double r203834 = r203820 - r203829;
        double r203835 = r203833 * r203834;
        double r203836 = r203820 / r203835;
        double r203837 = 5.0;
        double r203838 = r203837 * r203829;
        double r203839 = r203824 * r203821;
        double r203840 = r203822 * r203831;
        double r203841 = sqrt(r203840);
        double r203842 = r203839 * r203841;
        double r203843 = r203842 * r203834;
        double r203844 = r203838 / r203843;
        double r203845 = r203836 - r203844;
        return r203845;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
  2. Using strategy rm

  3. Applied div-sub0.4

    \[\leadsto \color{blue}{\frac{1}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}}\]
  4. Using strategy rm

  5. Applied sqrt-prod0.4

    \[\leadsto \frac{1}{\left(\left(\pi \cdot t\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  6. Applied associate-*r*0.4

    \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)} \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  7. Simplified0.4

    \[\leadsto \frac{1}{\left(\color{blue}{\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  8. Final simplification0.4

    \[\leadsto \frac{1}{\left(\left(t \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))