Average Error: 0.4 → 0.3
Time: 2.0m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{t \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{1 - v \cdot v}\]
\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{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{t \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{1 - v \cdot v}
double f(double v, double t) {
        double r31245030 = 1.0;
        double r31245031 = 5.0;
        double r31245032 = v;
        double r31245033 = r31245032 * r31245032;
        double r31245034 = r31245031 * r31245033;
        double r31245035 = r31245030 - r31245034;
        double r31245036 = atan2(1.0, 0.0);
        double r31245037 = t;
        double r31245038 = r31245036 * r31245037;
        double r31245039 = 2.0;
        double r31245040 = 3.0;
        double r31245041 = r31245040 * r31245033;
        double r31245042 = r31245030 - r31245041;
        double r31245043 = r31245039 * r31245042;
        double r31245044 = sqrt(r31245043);
        double r31245045 = r31245038 * r31245044;
        double r31245046 = r31245030 - r31245033;
        double r31245047 = r31245045 * r31245046;
        double r31245048 = r31245035 / r31245047;
        return r31245048;
}
↓
double f(double v, double t) {
        double r31245049 = 1.0;
        double r31245050 = v;
        double r31245051 = r31245050 * r31245050;
        double r31245052 = 5.0;
        double r31245053 = r31245051 * r31245052;
        double r31245054 = r31245049 - r31245053;
        double r31245055 = atan2(1.0, 0.0);
        double r31245056 = r31245054 / r31245055;
        double r31245057 = t;
        double r31245058 = 2.0;
        double r31245059 = 6.0;
        double r31245060 = r31245051 * r31245059;
        double r31245061 = r31245058 - r31245060;
        double r31245062 = sqrt(r31245061);
        double r31245063 = r31245057 * r31245062;
        double r31245064 = r31245056 / r31245063;
        double r31245065 = r31245049 - r31245051;
        double r31245066 = r31245064 / r31245065;
        return r31245066;
}

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 associate-/r*0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied times-frac0.5

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

  8. Applied associate-/r*0.3

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

  10. Applied frac-times0.3

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

  11. Simplified0.3

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

  12. Simplified0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\color{blue}{\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot t}}}{1 - v \cdot v}\]

  13. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{t \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{1 - v \cdot v}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))