Average Error: 0.5 → 0.4
Time: 7.7m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\sqrt{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{1 - v \cdot v}}{\sqrt{2 \cdot (\left(-v \cdot v\right) \cdot 3 + 1)_*}}}}{t} \cdot \frac{\sqrt{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{1 - v \cdot v}}{\sqrt{2 \cdot (\left(-v \cdot v\right) \cdot 3 + 1)_*}}}}{\pi}\]
double f(double v, double t) {
        double r47319460 = 1.0;
        double r47319461 = 5.0;
        double r47319462 = v;
        double r47319463 = r47319462 * r47319462;
        double r47319464 = r47319461 * r47319463;
        double r47319465 = r47319460 - r47319464;
        double r47319466 = atan2(1.0, 0.0);
        double r47319467 = t;
        double r47319468 = r47319466 * r47319467;
        double r47319469 = 2.0;
        double r47319470 = 3.0;
        double r47319471 = r47319470 * r47319463;
        double r47319472 = r47319460 - r47319471;
        double r47319473 = r47319469 * r47319472;
        double r47319474 = sqrt(r47319473);
        double r47319475 = r47319468 * r47319474;
        double r47319476 = r47319460 - r47319463;
        double r47319477 = r47319475 * r47319476;
        double r47319478 = r47319465 / r47319477;
        return r47319478;
}
↓
double f(double v, double t) {
        double r47319479 = -5.0;
        double r47319480 = v;
        double r47319481 = r47319480 * r47319480;
        double r47319482 = 1.0;
        double r47319483 = fma(r47319479, r47319481, r47319482);
        double r47319484 = r47319482 - r47319481;
        double r47319485 = r47319483 / r47319484;
        double r47319486 = 2.0;
        double r47319487 = -r47319481;
        double r47319488 = 3.0;
        double r47319489 = fma(r47319487, r47319488, r47319482);
        double r47319490 = r47319486 * r47319489;
        double r47319491 = sqrt(r47319490);
        double r47319492 = r47319485 / r47319491;
        double r47319493 = sqrt(r47319492);
        double r47319494 = t;
        double r47319495 = r47319493 / r47319494;
        double r47319496 = atan2(1.0, 0.0);
        double r47319497 = r47319493 / r47319496;
        double r47319498 = r47319495 * r47319497;
        return r47319498;
}


\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{\sqrt{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{1 - v \cdot v}}{\sqrt{2 \cdot (\left(-v \cdot v\right) \cdot 3 + 1)_*}}}}{t} \cdot \frac{\sqrt{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{1 - v \cdot v}}{\sqrt{2 \cdot (\left(-v \cdot v\right) \cdot 3 + 1)_*}}}}{\pi}

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  4. Applied associate-/r*0.4

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied times-frac0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))