Average Error: 1.0 → 0.2
Time: 5.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \mathsf{fma}\left(\sqrt{2}, 1, 3 \cdot \frac{{v}^{4}}{\sqrt{2}} - \mathsf{fma}\left(3, \frac{{v}^{2}}{\sqrt{2}}, \mathsf{fma}\left(4.5, \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{3}}, \sqrt{2} \cdot {v}^{2}\right)\right)\right)} \cdot \left(1 + v \cdot v\right)\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \mathsf{fma}\left(\sqrt{2}, 1, 3 \cdot \frac{{v}^{4}}{\sqrt{2}} - \mathsf{fma}\left(3, \frac{{v}^{2}}{\sqrt{2}}, \mathsf{fma}\left(4.5, \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{3}}, \sqrt{2} \cdot {v}^{2}\right)\right)\right)} \cdot \left(1 + v \cdot v\right)
double f(double v) {
        double r251093 = 4.0;
        double r251094 = 3.0;
        double r251095 = atan2(1.0, 0.0);
        double r251096 = r251094 * r251095;
        double r251097 = 1.0;
        double r251098 = v;
        double r251099 = r251098 * r251098;
        double r251100 = r251097 - r251099;
        double r251101 = r251096 * r251100;
        double r251102 = 2.0;
        double r251103 = 6.0;
        double r251104 = r251103 * r251099;
        double r251105 = r251102 - r251104;
        double r251106 = sqrt(r251105);
        double r251107 = r251101 * r251106;
        double r251108 = r251093 / r251107;
        return r251108;
}
↓
double f(double v) {
        double r251109 = 4.0;
        double r251110 = 3.0;
        double r251111 = atan2(1.0, 0.0);
        double r251112 = r251110 * r251111;
        double r251113 = r251109 / r251112;
        double r251114 = v;
        double r251115 = 1.0;
        double r251116 = fma(r251114, r251114, r251115);
        double r251117 = 2.0;
        double r251118 = sqrt(r251117);
        double r251119 = 4.0;
        double r251120 = pow(r251114, r251119);
        double r251121 = r251120 / r251118;
        double r251122 = r251110 * r251121;
        double r251123 = 2.0;
        double r251124 = pow(r251114, r251123);
        double r251125 = r251124 / r251118;
        double r251126 = 4.5;
        double r251127 = 3.0;
        double r251128 = pow(r251118, r251127);
        double r251129 = r251120 / r251128;
        double r251130 = r251118 * r251124;
        double r251131 = fma(r251126, r251129, r251130);
        double r251132 = fma(r251110, r251125, r251131);
        double r251133 = r251122 - r251132;
        double r251134 = fma(r251118, r251115, r251133);
        double r251135 = r251116 * r251134;
        double r251136 = r251113 / r251135;
        double r251137 = r251114 * r251114;
        double r251138 = r251115 + r251137;
        double r251139 = r251136 * r251138;
        return r251139;
}

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied flip--1.0

    \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \color{blue}{\frac{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}{1 + v \cdot v}}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  4. Applied associate-*r/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 + v \cdot v}} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  5. Applied associate-*l/1.0

    \[\leadsto \frac{4}{\color{blue}{\frac{\left(\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}{1 + v \cdot v}}}\]

  6. Applied associate-/r/1.0

    \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)}\]

  7. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}} \cdot \left(1 + v \cdot v\right)\]

  8. Taylor expanded around 0 0.2

    \[\leadsto \frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \color{blue}{\left(\left(1 \cdot \sqrt{2} + 3 \cdot \frac{{v}^{4}}{\sqrt{2}}\right) - \left(3 \cdot \frac{{v}^{2}}{\sqrt{2}} + \left(4.5 \cdot \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{3}} + \sqrt{2} \cdot {v}^{2}\right)\right)\right)}} \cdot \left(1 + v \cdot v\right)\]

  9. Simplified0.2

    \[\leadsto \frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{2}, 1, 3 \cdot \frac{{v}^{4}}{\sqrt{2}} - \mathsf{fma}\left(3, \frac{{v}^{2}}{\sqrt{2}}, \mathsf{fma}\left(4.5, \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{3}}, \sqrt{2} \cdot {v}^{2}\right)\right)\right)}} \cdot \left(1 + v \cdot v\right)\]

  10. Final simplification0.2

    \[\leadsto \frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \mathsf{fma}\left(\sqrt{2}, 1, 3 \cdot \frac{{v}^{4}}{\sqrt{2}} - \mathsf{fma}\left(3, \frac{{v}^{2}}{\sqrt{2}}, \mathsf{fma}\left(4.5, \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{3}}, \sqrt{2} \cdot {v}^{2}\right)\right)\right)} \cdot \left(1 + v \cdot v\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))