Average Error: 0.6 → 0.6
Time: 8.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\right)}
double f(double v) {
        double r576 = 1.0;
        double r577 = 5.0;
        double r578 = v;
        double r579 = r578 * r578;
        double r580 = r577 * r579;
        double r581 = r576 - r580;
        double r582 = r579 - r576;
        double r583 = r581 / r582;
        double r584 = acos(r583);
        return r584;
}
↓
double f(double v) {
        double r585 = 1.0;
        double r586 = r585 * r585;
        double r587 = 5.0;
        double r588 = v;
        double r589 = r588 * r588;
        double r590 = r587 * r589;
        double r591 = r590 * r590;
        double r592 = r586 - r591;
        double r593 = r589 - r585;
        double r594 = r585 + r590;
        double r595 = r593 * r594;
        double r596 = r592 / r595;
        double r597 = acos(r596);
        double r598 = log(r597);
        double r599 = exp(r598);
        return r599;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied flip--0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{1 + 5 \cdot \left(v \cdot v\right)}}}{v \cdot v - 1}\right)\]

  4. Applied associate-/l/0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)}\]
  5. Using strategy rm

  6. Applied add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\right)}}\]

  7. Final simplification0.6

    \[\leadsto e^{\log \left(\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\right)}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))