Average Error: 0.5 → 0.5
Time: 7.1s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot \left(t \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\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 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot \left(t \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r112955 = 1.0;
        double r112956 = 5.0;
        double r112957 = v;
        double r112958 = r112957 * r112957;
        double r112959 = r112956 * r112958;
        double r112960 = r112955 - r112959;
        double r112961 = atan2(1.0, 0.0);
        double r112962 = t;
        double r112963 = r112961 * r112962;
        double r112964 = 2.0;
        double r112965 = 3.0;
        double r112966 = r112965 * r112958;
        double r112967 = r112955 - r112966;
        double r112968 = r112964 * r112967;
        double r112969 = sqrt(r112968);
        double r112970 = r112963 * r112969;
        double r112971 = r112955 - r112958;
        double r112972 = r112970 * r112971;
        double r112973 = r112960 / r112972;
        return r112973;
}
↓
double f(double v, double t) {
        double r112974 = 1.0;
        double r112975 = 5.0;
        double r112976 = v;
        double r112977 = r112976 * r112976;
        double r112978 = r112975 * r112977;
        double r112979 = r112974 - r112978;
        double r112980 = atan2(1.0, 0.0);
        double r112981 = t;
        double r112982 = 2.0;
        double r112983 = sqrt(r112982);
        double r112984 = r112981 * r112983;
        double r112985 = r112980 * r112984;
        double r112986 = 3.0;
        double r112987 = r112986 * r112977;
        double r112988 = r112974 - r112987;
        double r112989 = sqrt(r112988);
        double r112990 = r112985 * r112989;
        double r112991 = r112974 - r112977;
        double r112992 = r112990 * r112991;
        double r112993 = r112979 / r112992;
        return r112993;
}

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 sqrt-prod0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied associate-*l*0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 +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)))))