Average Error: 0.4 → 0.1
Time: 17.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\left|\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right|}{\pi} \cdot \frac{\sqrt{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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{\left|\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right|}{\pi} \cdot \frac{\sqrt{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r457 = 1.0;
        double r458 = 5.0;
        double r459 = v;
        double r460 = r459 * r459;
        double r461 = r458 * r460;
        double r462 = r457 - r461;
        double r463 = atan2(1.0, 0.0);
        double r464 = t;
        double r465 = r463 * r464;
        double r466 = 2.0;
        double r467 = 3.0;
        double r468 = r467 * r460;
        double r469 = r457 - r468;
        double r470 = r466 * r469;
        double r471 = sqrt(r470);
        double r472 = r465 * r471;
        double r473 = r457 - r460;
        double r474 = r472 * r473;
        double r475 = r462 / r474;
        return r475;
}
↓
double f(double v, double t) {
        double r476 = 1.0;
        double r477 = 5.0;
        double r478 = v;
        double r479 = r478 * r478;
        double r480 = r477 * r479;
        double r481 = r476 - r480;
        double r482 = cbrt(r481);
        double r483 = fabs(r482);
        double r484 = atan2(1.0, 0.0);
        double r485 = r483 / r484;
        double r486 = sqrt(r482);
        double r487 = 2.0;
        double r488 = 3.0;
        double r489 = r488 * r479;
        double r490 = r476 - r489;
        double r491 = r487 * r490;
        double r492 = sqrt(r491);
        double r493 = r486 / r492;
        double r494 = r485 * r493;
        double r495 = t;
        double r496 = r494 / r495;
        double r497 = sqrt(r481);
        double r498 = r476 - r479;
        double r499 = r497 / r498;
        double r500 = r496 * r499;
        return r500;
}

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 add-sqr-sqrt0.5

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

  4. Applied times-frac0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied sqrt-prod0.5

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

  8. Applied times-frac0.5

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

  9. Simplified0.3

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

  11. Applied associate-*l/0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 +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)))))