Average Error: 1.0 → 0.0
Time: 21.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{\frac{\frac{\pi \cdot \left(3 \cdot \left(\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{4}{\frac{\frac{\pi \cdot \left(3 \cdot \left(\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}
double f(double v) {
        double r184042 = 4.0;
        double r184043 = 3.0;
        double r184044 = atan2(1.0, 0.0);
        double r184045 = r184043 * r184044;
        double r184046 = 1.0;
        double r184047 = v;
        double r184048 = r184047 * r184047;
        double r184049 = r184046 - r184048;
        double r184050 = r184045 * r184049;
        double r184051 = 2.0;
        double r184052 = 6.0;
        double r184053 = r184052 * r184048;
        double r184054 = r184051 - r184053;
        double r184055 = sqrt(r184054);
        double r184056 = r184050 * r184055;
        double r184057 = r184042 / r184056;
        return r184057;
}
↓
double f(double v) {
        double r184058 = 4.0;
        double r184059 = atan2(1.0, 0.0);
        double r184060 = 3.0;
        double r184061 = 2.0;
        double r184062 = 3.0;
        double r184063 = pow(r184061, r184062);
        double r184064 = 6.0;
        double r184065 = v;
        double r184066 = r184065 * r184065;
        double r184067 = r184064 * r184066;
        double r184068 = pow(r184067, r184062);
        double r184069 = r184063 - r184068;
        double r184070 = sqrt(r184069);
        double r184071 = 1.0;
        double r184072 = pow(r184071, r184062);
        double r184073 = 6.0;
        double r184074 = pow(r184065, r184073);
        double r184075 = r184072 - r184074;
        double r184076 = r184070 * r184075;
        double r184077 = r184060 * r184076;
        double r184078 = r184059 * r184077;
        double r184079 = r184061 * r184061;
        double r184080 = r184067 * r184067;
        double r184081 = r184061 * r184067;
        double r184082 = r184080 + r184081;
        double r184083 = r184079 + r184082;
        double r184084 = sqrt(r184083);
        double r184085 = r184078 / r184084;
        double r184086 = r184071 * r184071;
        double r184087 = r184066 * r184066;
        double r184088 = r184071 * r184066;
        double r184089 = r184087 + r184088;
        double r184090 = r184086 + r184089;
        double r184091 = r184085 / r184090;
        double r184092 = r184058 / r184091;
        return r184092;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied flip3--1.0

    \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  4. Applied associate-*r/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  5. Applied associate-*l/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}\]

  6. Simplified0.0

    \[\leadsto \frac{4}{\frac{\color{blue}{\pi \cdot \left(3 \cdot \left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]
  7. Using strategy rm

  8. Applied flip3--0.0

    \[\leadsto \frac{4}{\frac{\pi \cdot \left(3 \cdot \left(\sqrt{\color{blue}{\frac{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  9. Applied sqrt-div0.0

    \[\leadsto \frac{4}{\frac{\pi \cdot \left(3 \cdot \left(\color{blue}{\frac{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  10. Applied associate-*l/0.0

    \[\leadsto \frac{4}{\frac{\pi \cdot \left(3 \cdot \color{blue}{\frac{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  11. Applied associate-*r/0.0

    \[\leadsto \frac{4}{\frac{\pi \cdot \color{blue}{\frac{3 \cdot \left(\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)\right)}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  12. Applied associate-*r/0.0

    \[\leadsto \frac{4}{\frac{\color{blue}{\frac{\pi \cdot \left(3 \cdot \left(\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  13. Final simplification0.0

    \[\leadsto \frac{4}{\frac{\frac{\pi \cdot \left(3 \cdot \left(\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)\right)\right)}{\sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))