Average Error: 0.4 → 0.1
Time: 1.8m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)}}}{\left(1 - v \cdot v\right) \cdot t}\]
\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(-5, \left(v \cdot v\right), 1\right)}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)}}}{\left(1 - v \cdot v\right) \cdot t}
double f(double v, double t) {
        double r13269691 = 1.0;
        double r13269692 = 5.0;
        double r13269693 = v;
        double r13269694 = r13269693 * r13269693;
        double r13269695 = r13269692 * r13269694;
        double r13269696 = r13269691 - r13269695;
        double r13269697 = atan2(1.0, 0.0);
        double r13269698 = t;
        double r13269699 = r13269697 * r13269698;
        double r13269700 = 2.0;
        double r13269701 = 3.0;
        double r13269702 = r13269701 * r13269694;
        double r13269703 = r13269691 - r13269702;
        double r13269704 = r13269700 * r13269703;
        double r13269705 = sqrt(r13269704);
        double r13269706 = r13269699 * r13269705;
        double r13269707 = r13269691 - r13269694;
        double r13269708 = r13269706 * r13269707;
        double r13269709 = r13269696 / r13269708;
        return r13269709;
}
↓
double f(double v, double t) {
        double r13269710 = -5.0;
        double r13269711 = v;
        double r13269712 = r13269711 * r13269711;
        double r13269713 = 1.0;
        double r13269714 = fma(r13269710, r13269712, r13269713);
        double r13269715 = atan2(1.0, 0.0);
        double r13269716 = r13269714 / r13269715;
        double r13269717 = -6.0;
        double r13269718 = 2.0;
        double r13269719 = fma(r13269717, r13269712, r13269718);
        double r13269720 = sqrt(r13269719);
        double r13269721 = r13269716 / r13269720;
        double r13269722 = r13269713 - r13269712;
        double r13269723 = t;
        double r13269724 = r13269722 * r13269723;
        double r13269725 = r13269721 / r13269724;
        return r13269725;
}

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. Simplified0.3

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

  4. Applied div-inv0.4

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

  5. Applied associate-/l*0.4

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

  6. Simplified0.3

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

  8. Applied associate-/r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019132 +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)))))