Average Error: 0.5 → 0.6
Time: 13.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r215706 = 1.0;
        double r215707 = 5.0;
        double r215708 = v;
        double r215709 = r215708 * r215708;
        double r215710 = r215707 * r215709;
        double r215711 = r215706 - r215710;
        double r215712 = r215709 - r215706;
        double r215713 = r215711 / r215712;
        double r215714 = acos(r215713);
        return r215714;
}
↓
double f(double v) {
        double r215715 = 1.0;
        double r215716 = 5.0;
        double r215717 = v;
        double r215718 = r215717 * r215717;
        double r215719 = r215716 * r215718;
        double r215720 = r215715 - r215719;
        double r215721 = sqrt(r215720);
        double r215722 = r215718 - r215715;
        double r215723 = r215721 / r215722;
        double r215724 = r215721 * r215723;
        double r215725 = acos(r215724);
        return r215725;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  3. Applied *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied times-frac0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))