Average Error: 1.0 → 0.0
Time: 18.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(\pi \cdot 1\right) \cdot 3 + \left(-3 \cdot \left({v}^{2} \cdot \pi\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\frac{4}{\left(\pi \cdot 1\right) \cdot 3 + \left(-3 \cdot \left({v}^{2} \cdot \pi\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r222109 = 4.0;
        double r222110 = 3.0;
        double r222111 = atan2(1.0, 0.0);
        double r222112 = r222110 * r222111;
        double r222113 = 1.0;
        double r222114 = v;
        double r222115 = r222114 * r222114;
        double r222116 = r222113 - r222115;
        double r222117 = r222112 * r222116;
        double r222118 = 2.0;
        double r222119 = 6.0;
        double r222120 = r222119 * r222115;
        double r222121 = r222118 - r222120;
        double r222122 = sqrt(r222121);
        double r222123 = r222117 * r222122;
        double r222124 = r222109 / r222123;
        return r222124;
}
↓
double f(double v) {
        double r222125 = 4.0;
        double r222126 = atan2(1.0, 0.0);
        double r222127 = 1.0;
        double r222128 = r222126 * r222127;
        double r222129 = 3.0;
        double r222130 = r222128 * r222129;
        double r222131 = v;
        double r222132 = 2.0;
        double r222133 = pow(r222131, r222132);
        double r222134 = r222133 * r222126;
        double r222135 = r222129 * r222134;
        double r222136 = -r222135;
        double r222137 = r222130 + r222136;
        double r222138 = r222125 / r222137;
        double r222139 = 2.0;
        double r222140 = 6.0;
        double r222141 = r222131 * r222131;
        double r222142 = r222140 * r222141;
        double r222143 = r222139 - r222142;
        double r222144 = sqrt(r222143);
        double r222145 = r222138 / r222144;
        return r222145;
}

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 associate-/r*0.0

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

  4. Simplified0.0

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

  6. Applied sub-neg0.0

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

  7. Applied distribute-lft-in0.0

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

  8. Applied distribute-lft-in0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019305 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))