Average Error: 0.5 → 0.6
Time: 16.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)
double f(double v) {
        double r301435 = 1.0;
        double r301436 = 5.0;
        double r301437 = v;
        double r301438 = r301437 * r301437;
        double r301439 = r301436 * r301438;
        double r301440 = r301435 - r301439;
        double r301441 = r301438 - r301435;
        double r301442 = r301440 / r301441;
        double r301443 = acos(r301442);
        return r301443;
}
↓
double f(double v) {
        double r301444 = 1.0;
        double r301445 = v;
        double r301446 = r301445 * r301445;
        double r301447 = 1.0;
        double r301448 = r301446 - r301447;
        double r301449 = 5.0;
        double r301450 = r301449 * r301446;
        double r301451 = r301447 - r301450;
        double r301452 = r301448 / r301451;
        double r301453 = r301444 / r301452;
        double r301454 = acos(r301453);
        return r301454;
}

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 clear-num0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019351 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))