Average Error: 0.4 → 0.1
Time: 34.0s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{\mathsf{fma}\left(5 \cdot \left(-v\right), v, 1\right)}{{\left(\sqrt{\mathsf{fma}\left(-v, v \cdot 3, 1\right) \cdot 2}\right)}^{3}}}}{\mathsf{fma}\left(v, -v, 1\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(5 \cdot \left(-v\right), v, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(5 \cdot \left(-v\right), v, 1\right)}}{\pi}}{t}\]
\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{\sqrt[3]{\frac{\mathsf{fma}\left(5 \cdot \left(-v\right), v, 1\right)}{{\left(\sqrt{\mathsf{fma}\left(-v, v \cdot 3, 1\right) \cdot 2}\right)}^{3}}}}{\mathsf{fma}\left(v, -v, 1\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(5 \cdot \left(-v\right), v, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(5 \cdot \left(-v\right), v, 1\right)}}{\pi}}{t}
double f(double v, double t) {
        double r292092 = 1.0;
        double r292093 = 5.0;
        double r292094 = v;
        double r292095 = r292094 * r292094;
        double r292096 = r292093 * r292095;
        double r292097 = r292092 - r292096;
        double r292098 = atan2(1.0, 0.0);
        double r292099 = t;
        double r292100 = r292098 * r292099;
        double r292101 = 2.0;
        double r292102 = 3.0;
        double r292103 = r292102 * r292095;
        double r292104 = r292092 - r292103;
        double r292105 = r292101 * r292104;
        double r292106 = sqrt(r292105);
        double r292107 = r292100 * r292106;
        double r292108 = r292092 - r292095;
        double r292109 = r292107 * r292108;
        double r292110 = r292097 / r292109;
        return r292110;
}
↓
double f(double v, double t) {
        double r292111 = 5.0;
        double r292112 = v;
        double r292113 = -r292112;
        double r292114 = r292111 * r292113;
        double r292115 = 1.0;
        double r292116 = fma(r292114, r292112, r292115);
        double r292117 = 3.0;
        double r292118 = r292112 * r292117;
        double r292119 = fma(r292113, r292118, r292115);
        double r292120 = 2.0;
        double r292121 = r292119 * r292120;
        double r292122 = sqrt(r292121);
        double r292123 = 3.0;
        double r292124 = pow(r292122, r292123);
        double r292125 = r292116 / r292124;
        double r292126 = cbrt(r292125);
        double r292127 = fma(r292112, r292113, r292115);
        double r292128 = r292126 / r292127;
        double r292129 = cbrt(r292116);
        double r292130 = r292129 * r292129;
        double r292131 = atan2(1.0, 0.0);
        double r292132 = r292130 / r292131;
        double r292133 = r292128 * r292132;
        double r292134 = t;
        double r292135 = r292133 / r292134;
        return r292135;
}

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

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied times-frac0.5

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied associate-*r/0.3

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

  12. Applied associate-*l/0.1

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

  13. Simplified0.1

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

  15. Applied add-cbrt-cube0.9

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

  16. Applied cbrt-undiv0.1

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

  17. Simplified0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))