Average Error: 1.0 → 0.0
Time: 6.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}\]
\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{4}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}
double f(double v) {
        double r326548 = 4.0;
        double r326549 = 3.0;
        double r326550 = atan2(1.0, 0.0);
        double r326551 = r326549 * r326550;
        double r326552 = 1.0;
        double r326553 = v;
        double r326554 = r326553 * r326553;
        double r326555 = r326552 - r326554;
        double r326556 = r326551 * r326555;
        double r326557 = 2.0;
        double r326558 = 6.0;
        double r326559 = r326558 * r326554;
        double r326560 = r326557 - r326559;
        double r326561 = sqrt(r326560);
        double r326562 = r326556 * r326561;
        double r326563 = r326548 / r326562;
        return r326563;
}
↓
double f(double v) {
        double r326564 = 4.0;
        double r326565 = 3.0;
        double r326566 = atan2(1.0, 0.0);
        double r326567 = r326565 * r326566;
        double r326568 = 1.0;
        double r326569 = v;
        double r326570 = r326569 * r326569;
        double r326571 = r326568 - r326570;
        double r326572 = r326567 * r326571;
        double r326573 = 2.0;
        double r326574 = 6.0;
        double r326575 = r326574 * r326570;
        double r326576 = r326573 - r326575;
        double r326577 = sqrt(r326576);
        double r326578 = r326572 * r326577;
        double r326579 = log(r326578);
        double r326580 = exp(r326579);
        double r326581 = r326564 / r326580;
        return r326581;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 add-exp-log1.0

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

  4. Applied add-exp-log1.0

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

  5. Applied add-exp-log1.0

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

  6. Applied add-exp-log1.0

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

  7. Applied prod-exp1.0

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

  8. Applied prod-exp1.0

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

  9. Applied prod-exp0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

    \[\leadsto \frac{4}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))