Average Error: 0.4 → 0.3
Time: 4.5m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}} \cdot \frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\sqrt{2}}}{t}\]
\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{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}} \cdot \frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\sqrt{2}}}{t}
double f(double v, double t) {
        double r65792590 = 1.0;
        double r65792591 = 5.0;
        double r65792592 = v;
        double r65792593 = r65792592 * r65792592;
        double r65792594 = r65792591 * r65792593;
        double r65792595 = r65792590 - r65792594;
        double r65792596 = atan2(1.0, 0.0);
        double r65792597 = t;
        double r65792598 = r65792596 * r65792597;
        double r65792599 = 2.0;
        double r65792600 = 3.0;
        double r65792601 = r65792600 * r65792593;
        double r65792602 = r65792590 - r65792601;
        double r65792603 = r65792599 * r65792602;
        double r65792604 = sqrt(r65792603);
        double r65792605 = r65792598 * r65792604;
        double r65792606 = r65792590 - r65792593;
        double r65792607 = r65792605 * r65792606;
        double r65792608 = r65792595 / r65792607;
        return r65792608;
}
↓
double f(double v, double t) {
        double r65792609 = 1.0;
        double r65792610 = atan2(1.0, 0.0);
        double r65792611 = r65792609 / r65792610;
        double r65792612 = v;
        double r65792613 = r65792612 * r65792612;
        double r65792614 = r65792609 - r65792613;
        double r65792615 = r65792611 / r65792614;
        double r65792616 = -3.0;
        double r65792617 = fma(r65792613, r65792616, r65792609);
        double r65792618 = sqrt(r65792617);
        double r65792619 = r65792615 / r65792618;
        double r65792620 = -5.0;
        double r65792621 = fma(r65792620, r65792613, r65792609);
        double r65792622 = 2.0;
        double r65792623 = sqrt(r65792622);
        double r65792624 = r65792621 / r65792623;
        double r65792625 = t;
        double r65792626 = r65792624 / r65792625;
        double r65792627 = r65792619 * r65792626;
        return r65792627;
}

Error

Bits error versus v

Bits error versus t

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. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{1 - v \cdot v}}{\sqrt{2 \cdot (\left(-v \cdot v\right) \cdot 3 + 1)_*} \cdot \left(t \cdot \pi\right)}}\]
  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 sqrt-prod0.4

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

  7. Applied div-inv0.4

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

  8. Applied times-frac0.4

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019112 +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)))))