Average Error: 0.4 → 0.1
Time: 17.8s
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 r490 = 1.0;
        double r491 = 5.0;
        double r492 = v;
        double r493 = r492 * r492;
        double r494 = r491 * r493;
        double r495 = r490 - r494;
        double r496 = atan2(1.0, 0.0);
        double r497 = t;
        double r498 = r496 * r497;
        double r499 = 2.0;
        double r500 = 3.0;
        double r501 = r500 * r493;
        double r502 = r490 - r501;
        double r503 = r499 * r502;
        double r504 = sqrt(r503);
        double r505 = r498 * r504;
        double r506 = r490 - r493;
        double r507 = r505 * r506;
        double r508 = r495 / r507;
        return r508;
}
↓
double f(double v, double t) {
        double r509 = 1.0;
        double r510 = 5.0;
        double r511 = v;
        double r512 = r511 * r511;
        double r513 = r510 * r512;
        double r514 = r509 - r513;
        double r515 = cbrt(r514);
        double r516 = fabs(r515);
        double r517 = atan2(1.0, 0.0);
        double r518 = r516 / r517;
        double r519 = sqrt(r515);
        double r520 = 2.0;
        double r521 = 3.0;
        double r522 = r521 * r512;
        double r523 = r509 - r522;
        double r524 = r520 * r523;
        double r525 = sqrt(r524);
        double r526 = r519 / r525;
        double r527 = r518 * r526;
        double r528 = t;
        double r529 = r527 / r528;
        double r530 = sqrt(r514);
        double r531 = r509 - r512;
        double r532 = r530 / r531;
        double r533 = r529 * r532;
        return r533;
}

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 
(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)))))