Average Error: 0.0 → 0.0
Time: 16.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2}\right)}{4}\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
e^{\log \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2}\right)}{4}\right)}
double f(double v) {
        double r6749086 = 2.0;
        double r6749087 = sqrt(r6749086);
        double r6749088 = 4.0;
        double r6749089 = r6749087 / r6749088;
        double r6749090 = 1.0;
        double r6749091 = 3.0;
        double r6749092 = v;
        double r6749093 = r6749092 * r6749092;
        double r6749094 = r6749091 * r6749093;
        double r6749095 = r6749090 - r6749094;
        double r6749096 = sqrt(r6749095);
        double r6749097 = r6749089 * r6749096;
        double r6749098 = r6749090 - r6749093;
        double r6749099 = r6749097 * r6749098;
        return r6749099;
}
↓
double f(double v) {
        double r6749100 = 1.0;
        double r6749101 = 3.0;
        double r6749102 = v;
        double r6749103 = r6749102 * r6749102;
        double r6749104 = r6749101 * r6749103;
        double r6749105 = r6749100 - r6749104;
        double r6749106 = sqrt(r6749105);
        double r6749107 = r6749100 - r6749103;
        double r6749108 = 2.0;
        double r6749109 = sqrt(r6749108);
        double r6749110 = r6749107 * r6749109;
        double r6749111 = r6749106 * r6749110;
        double r6749112 = 4.0;
        double r6749113 = r6749111 / r6749112;
        double r6749114 = log(r6749113);
        double r6749115 = exp(r6749114);
        return r6749115;
}

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-exp-log0.0

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

  4. Applied add-exp-log0.0

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

  5. Applied add-exp-log0.0

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

  6. Applied add-exp-log0.0

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

  7. Applied div-exp0.0

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

  8. Applied prod-exp0.0

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

  9. Applied prod-exp0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))