Average Error: 0.0 → 0.0
Time: 19.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\]
\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(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}
double f(double v) {
        double r147423 = 2.0;
        double r147424 = sqrt(r147423);
        double r147425 = 4.0;
        double r147426 = r147424 / r147425;
        double r147427 = 1.0;
        double r147428 = 3.0;
        double r147429 = v;
        double r147430 = r147429 * r147429;
        double r147431 = r147428 * r147430;
        double r147432 = r147427 - r147431;
        double r147433 = sqrt(r147432);
        double r147434 = r147426 * r147433;
        double r147435 = r147427 - r147430;
        double r147436 = r147434 * r147435;
        return r147436;
}
↓
double f(double v) {
        double r147437 = 2.0;
        double r147438 = sqrt(r147437);
        double r147439 = 4.0;
        double r147440 = r147438 / r147439;
        double r147441 = 1.0;
        double r147442 = 3.0;
        double r147443 = v;
        double r147444 = r147443 * r147443;
        double r147445 = r147442 * r147444;
        double r147446 = r147441 - r147445;
        double r147447 = sqrt(r147446);
        double r147448 = r147440 * r147447;
        double r147449 = r147441 - r147444;
        double r147450 = r147448 * r147449;
        double r147451 = log(r147450);
        double r147452 = exp(r147451);
        return r147452;
}

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

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))