Average Error: 0.5 → 0.5
Time: 32.4s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\frac{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}\right)}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}\]
\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 - 5 \cdot \left(v \cdot v\right)}{\frac{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}\right)}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r185946 = 1.0;
        double r185947 = 5.0;
        double r185948 = v;
        double r185949 = r185948 * r185948;
        double r185950 = r185947 * r185949;
        double r185951 = r185946 - r185950;
        double r185952 = atan2(1.0, 0.0);
        double r185953 = t;
        double r185954 = r185952 * r185953;
        double r185955 = 2.0;
        double r185956 = 3.0;
        double r185957 = r185956 * r185949;
        double r185958 = r185946 - r185957;
        double r185959 = r185955 * r185958;
        double r185960 = sqrt(r185959);
        double r185961 = r185954 * r185960;
        double r185962 = r185946 - r185949;
        double r185963 = r185961 * r185962;
        double r185964 = r185951 / r185963;
        return r185964;
}
↓
double f(double v, double t) {
        double r185965 = 1.0;
        double r185966 = 5.0;
        double r185967 = v;
        double r185968 = r185967 * r185967;
        double r185969 = r185966 * r185968;
        double r185970 = r185965 - r185969;
        double r185971 = atan2(1.0, 0.0);
        double r185972 = t;
        double r185973 = 2.0;
        double r185974 = r185965 * r185965;
        double r185975 = 3.0;
        double r185976 = r185975 * r185975;
        double r185977 = 4.0;
        double r185978 = pow(r185967, r185977);
        double r185979 = r185976 * r185978;
        double r185980 = r185974 - r185979;
        double r185981 = r185973 * r185980;
        double r185982 = sqrt(r185981);
        double r185983 = r185972 * r185982;
        double r185984 = r185971 * r185983;
        double r185985 = r185975 * r185968;
        double r185986 = r185965 + r185985;
        double r185987 = sqrt(r185986);
        double r185988 = r185984 / r185987;
        double r185989 = r185965 - r185968;
        double r185990 = r185988 * r185989;
        double r185991 = r185970 / r185990;
        return r185991;
}

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 0.5

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

  3. Applied flip--0.5

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

  4. Applied associate-*r/0.5

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

  5. Applied sqrt-div0.5

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

  6. Applied associate-*r/0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\frac{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}\right)}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))