Average Error: 0.5 → 0.5
Time: 15.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - 1 \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - 1 \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r178467 = 1.0;
        double r178468 = 5.0;
        double r178469 = v;
        double r178470 = r178469 * r178469;
        double r178471 = r178468 * r178470;
        double r178472 = r178467 - r178471;
        double r178473 = r178470 - r178467;
        double r178474 = r178472 / r178473;
        double r178475 = acos(r178474);
        return r178475;
}
↓
double f(double v) {
        double r178476 = 1.0;
        double r178477 = 1.0;
        double r178478 = 5.0;
        double r178479 = v;
        double r178480 = r178479 * r178479;
        double r178481 = r178478 * r178480;
        double r178482 = r178477 * r178481;
        double r178483 = r178476 - r178482;
        double r178484 = r178480 - r178476;
        double r178485 = r178483 / r178484;
        double r178486 = acos(r178485);
        return r178486;
}

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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019298 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))