Average Error: 0.5 → 0.5
Time: 4.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}
double f(double v) {
        double r232266 = 1.0;
        double r232267 = 5.0;
        double r232268 = v;
        double r232269 = r232268 * r232268;
        double r232270 = r232267 * r232269;
        double r232271 = r232266 - r232270;
        double r232272 = r232269 - r232266;
        double r232273 = r232271 / r232272;
        double r232274 = acos(r232273);
        return r232274;
}
↓
double f(double v) {
        double r232275 = atan2(1.0, 0.0);
        double r232276 = 2.0;
        double r232277 = r232275 / r232276;
        double r232278 = 1.0;
        double r232279 = v;
        double r232280 = pow(r232279, r232276);
        double r232281 = r232280 - r232278;
        double r232282 = r232278 / r232281;
        double r232283 = 5.0;
        double r232284 = r232279 * r232279;
        double r232285 = r232283 * r232284;
        double r232286 = r232284 - r232278;
        double r232287 = r232285 / r232286;
        double r232288 = r232282 - r232287;
        double r232289 = asin(r232288);
        double r232290 = 3.0;
        double r232291 = pow(r232289, r232290);
        double r232292 = cbrt(r232291);
        double r232293 = r232277 - r232292;
        return r232293;
}

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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied div-sub0.5

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\frac{1}{v \cdot v - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]

  6. Simplified0.5

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\color{blue}{\frac{1}{{v}^{2} - 1}} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.5

    \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt[3]{\left(\sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\]

  9. Simplified0.5

    \[\leadsto \frac{\pi}{2} - \sqrt[3]{\color{blue}{{\left(\sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}}\]

  10. Final simplification0.5

    \[\leadsto \frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))