Average Error: 1.0 → 0.0
Time: 7.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{4}}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\sqrt{4}}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r325448 = 4.0;
        double r325449 = 3.0;
        double r325450 = atan2(1.0, 0.0);
        double r325451 = r325449 * r325450;
        double r325452 = 1.0;
        double r325453 = v;
        double r325454 = r325453 * r325453;
        double r325455 = r325452 - r325454;
        double r325456 = r325451 * r325455;
        double r325457 = 2.0;
        double r325458 = 6.0;
        double r325459 = r325458 * r325454;
        double r325460 = r325457 - r325459;
        double r325461 = sqrt(r325460);
        double r325462 = r325456 * r325461;
        double r325463 = r325448 / r325462;
        return r325463;
}
↓
double f(double v) {
        double r325464 = 4.0;
        double r325465 = sqrt(r325464);
        double r325466 = 3.0;
        double r325467 = atan2(1.0, 0.0);
        double r325468 = 1.0;
        double r325469 = v;
        double r325470 = r325469 * r325469;
        double r325471 = r325468 - r325470;
        double r325472 = r325467 * r325471;
        double r325473 = r325466 * r325472;
        double r325474 = r325465 / r325473;
        double r325475 = 2.0;
        double r325476 = 6.0;
        double r325477 = r325476 * r325470;
        double r325478 = r325475 - r325477;
        double r325479 = sqrt(r325478);
        double r325480 = r325465 / r325479;
        double r325481 = r325474 * r325480;
        return r325481;
}

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 add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  6. Applied associate-*l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))