Average Error: 1.0 → 0.0
Time: 24.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{\frac{4}{3}} \cdot \frac{\sqrt{\frac{4}{3}}}{\pi - v \cdot \left(\pi \cdot v\right)}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]
\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{\sqrt{\frac{4}{3}} \cdot \frac{\sqrt{\frac{4}{3}}}{\pi - v \cdot \left(\pi \cdot v\right)}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}
double f(double v) {
        double r22548101 = 4.0;
        double r22548102 = 3.0;
        double r22548103 = atan2(1.0, 0.0);
        double r22548104 = r22548102 * r22548103;
        double r22548105 = 1.0;
        double r22548106 = v;
        double r22548107 = r22548106 * r22548106;
        double r22548108 = r22548105 - r22548107;
        double r22548109 = r22548104 * r22548108;
        double r22548110 = 2.0;
        double r22548111 = 6.0;
        double r22548112 = r22548111 * r22548107;
        double r22548113 = r22548110 - r22548112;
        double r22548114 = sqrt(r22548113);
        double r22548115 = r22548109 * r22548114;
        double r22548116 = r22548101 / r22548115;
        return r22548116;
}
↓
double f(double v) {
        double r22548117 = 1.3333333333333333;
        double r22548118 = sqrt(r22548117);
        double r22548119 = atan2(1.0, 0.0);
        double r22548120 = v;
        double r22548121 = r22548119 * r22548120;
        double r22548122 = r22548120 * r22548121;
        double r22548123 = r22548119 - r22548122;
        double r22548124 = r22548118 / r22548123;
        double r22548125 = r22548118 * r22548124;
        double r22548126 = -6.0;
        double r22548127 = r22548120 * r22548126;
        double r22548128 = 2.0;
        double r22548129 = fma(r22548127, r22548120, r22548128);
        double r22548130 = sqrt(r22548129);
        double r22548131 = r22548125 / r22548130;
        return r22548131;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))