Average Error: 0.5 → 0.1
Time: 28.5s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{\left(-\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(v \cdot v\right)\right) + \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{t}\]
\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{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{\left(-\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(v \cdot v\right)\right) + \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{t}
double f(double v, double t) {
        double r7194780 = 1.0;
        double r7194781 = 5.0;
        double r7194782 = v;
        double r7194783 = r7194782 * r7194782;
        double r7194784 = r7194781 * r7194783;
        double r7194785 = r7194780 - r7194784;
        double r7194786 = atan2(1.0, 0.0);
        double r7194787 = t;
        double r7194788 = r7194786 * r7194787;
        double r7194789 = 2.0;
        double r7194790 = 3.0;
        double r7194791 = r7194790 * r7194783;
        double r7194792 = r7194780 - r7194791;
        double r7194793 = r7194789 * r7194792;
        double r7194794 = sqrt(r7194793);
        double r7194795 = r7194788 * r7194794;
        double r7194796 = r7194780 - r7194783;
        double r7194797 = r7194795 * r7194796;
        double r7194798 = r7194785 / r7194797;
        return r7194798;
}
↓
double f(double v, double t) {
        double r7194799 = v;
        double r7194800 = r7194799 * r7194799;
        double r7194801 = -5.0;
        double r7194802 = 1.0;
        double r7194803 = fma(r7194800, r7194801, r7194802);
        double r7194804 = atan2(1.0, 0.0);
        double r7194805 = r7194803 / r7194804;
        double r7194806 = -6.0;
        double r7194807 = 2.0;
        double r7194808 = fma(r7194800, r7194806, r7194807);
        double r7194809 = sqrt(r7194808);
        double r7194810 = r7194809 * r7194800;
        double r7194811 = -r7194810;
        double r7194812 = r7194811 + r7194809;
        double r7194813 = r7194805 / r7194812;
        double r7194814 = t;
        double r7194815 = r7194813 / r7194814;
        return r7194815;
}

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. Simplified0.3

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

  4. Applied div-inv0.4

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi} \cdot \frac{1}{t}}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}\]

  5. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}, \left(-v\right) \cdot v, \sqrt{\mathsf{fma}\left(6, \left(-v\right) \cdot v, 2\right)}\right)}{\frac{1}{t}}}}\]

  6. Simplified0.3

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} + \left(-\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(v \cdot v\right)\right)\right) \cdot t}}\]
  7. Using strategy rm

  8. Applied associate-/r*0.1

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

  9. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{\left(-\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(v \cdot v\right)\right) + \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{t}\]

Reproduce

herbie shell --seed 2019163 +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)))))