Average Error: 0.0 → 0.0
Time: 18.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{1 \cdot 1 - {v}^{4}} \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}{\sqrt{1 + v \cdot v}} \cdot \sqrt{1 - v \cdot v}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{1 \cdot 1 - {v}^{4}} \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}{\sqrt{1 + v \cdot v}} \cdot \sqrt{1 - v \cdot v}
double f(double v) {
        double r168996 = 2.0;
        double r168997 = sqrt(r168996);
        double r168998 = 4.0;
        double r168999 = r168997 / r168998;
        double r169000 = 1.0;
        double r169001 = 3.0;
        double r169002 = v;
        double r169003 = r169002 * r169002;
        double r169004 = r169001 * r169003;
        double r169005 = r169000 - r169004;
        double r169006 = sqrt(r169005);
        double r169007 = r168999 * r169006;
        double r169008 = r169000 - r169003;
        double r169009 = r169007 * r169008;
        return r169009;
}
↓
double f(double v) {
        double r169010 = 1.0;
        double r169011 = r169010 * r169010;
        double r169012 = v;
        double r169013 = 4.0;
        double r169014 = pow(r169012, r169013);
        double r169015 = r169011 - r169014;
        double r169016 = sqrt(r169015);
        double r169017 = 2.0;
        double r169018 = sqrt(r169017);
        double r169019 = 4.0;
        double r169020 = r169018 / r169019;
        double r169021 = 3.0;
        double r169022 = r169012 * r169012;
        double r169023 = r169021 * r169022;
        double r169024 = r169010 - r169023;
        double r169025 = sqrt(r169024);
        double r169026 = r169020 * r169025;
        double r169027 = r169016 * r169026;
        double r169028 = r169010 + r169022;
        double r169029 = sqrt(r169028);
        double r169030 = r169027 / r169029;
        double r169031 = r169010 - r169022;
        double r169032 = sqrt(r169031);
        double r169033 = r169030 * r169032;
        return r169033;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-*r*0.0

    \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - v \cdot v}\right) \cdot \sqrt{1 - v \cdot v}}\]
  5. Using strategy rm

  6. Applied flip--0.0

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

  7. Applied sqrt-div0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))