Average Error: 0.5 → 0.1
Time: 3.2m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\pi \cdot \sqrt{(v \cdot \left(-6 \cdot v\right) + 2)_*}}}{\sqrt[3]{1 - v \cdot v}}}{\left(\sqrt[3]{1 - v \cdot v} \cdot \sqrt[3]{1 - v \cdot v}\right) \cdot t}\]
double f(double v, double t) {
        double r48578669 = 1.0;
        double r48578670 = 5.0;
        double r48578671 = v;
        double r48578672 = r48578671 * r48578671;
        double r48578673 = r48578670 * r48578672;
        double r48578674 = r48578669 - r48578673;
        double r48578675 = atan2(1.0, 0.0);
        double r48578676 = t;
        double r48578677 = r48578675 * r48578676;
        double r48578678 = 2.0;
        double r48578679 = 3.0;
        double r48578680 = r48578679 * r48578672;
        double r48578681 = r48578669 - r48578680;
        double r48578682 = r48578678 * r48578681;
        double r48578683 = sqrt(r48578682);
        double r48578684 = r48578677 * r48578683;
        double r48578685 = r48578669 - r48578672;
        double r48578686 = r48578684 * r48578685;
        double r48578687 = r48578674 / r48578686;
        return r48578687;
}
↓
double f(double v, double t) {
        double r48578688 = -5.0;
        double r48578689 = v;
        double r48578690 = r48578689 * r48578689;
        double r48578691 = 1.0;
        double r48578692 = fma(r48578688, r48578690, r48578691);
        double r48578693 = atan2(1.0, 0.0);
        double r48578694 = -6.0;
        double r48578695 = r48578694 * r48578689;
        double r48578696 = 2.0;
        double r48578697 = fma(r48578689, r48578695, r48578696);
        double r48578698 = sqrt(r48578697);
        double r48578699 = r48578693 * r48578698;
        double r48578700 = r48578692 / r48578699;
        double r48578701 = r48578691 - r48578690;
        double r48578702 = cbrt(r48578701);
        double r48578703 = r48578700 / r48578702;
        double r48578704 = r48578702 * r48578702;
        double r48578705 = t;
        double r48578706 = r48578704 * r48578705;
        double r48578707 = r48578703 / r48578706;
        return r48578707;
}


\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{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\pi \cdot \sqrt{(v \cdot \left(-6 \cdot v\right) + 2)_*}}}{\sqrt[3]{1 - v \cdot v}}}{\left(\sqrt[3]{1 - v \cdot v} \cdot \sqrt[3]{1 - v \cdot v}\right) \cdot t}

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*0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied *-un-lft-identity0.5

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

  8. Applied times-frac0.5

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

  9. Applied times-frac0.4

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

  10. Simplified0.3

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

  12. Applied associate-*l/0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}} \cdot \frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\sqrt[3]{1 - v \cdot v}}}{\pi}}{\left(\sqrt[3]{1 - v \cdot v} \cdot \sqrt[3]{1 - v \cdot v}\right) \cdot t}}\]

  13. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\pi \cdot \sqrt{(v \cdot \left(-6 \cdot v\right) + 2)_*}}}{\sqrt[3]{1 - v \cdot v}}}}{\left(\sqrt[3]{1 - v \cdot v} \cdot \sqrt[3]{1 - v \cdot v}\right) \cdot t}\]

  14. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{\pi \cdot \sqrt{(v \cdot \left(-6 \cdot v\right) + 2)_*}}}{\sqrt[3]{1 - v \cdot v}}}{\left(\sqrt[3]{1 - v \cdot v} \cdot \sqrt[3]{1 - v \cdot v}\right) \cdot t}\]

Reproduce

herbie shell --seed 2019101 +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)))))