Average Error: 0.6 → 0.8
Time: 16.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\frac{1}{8} \cdot {\pi}^{3} - {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\frac{1}{4} \cdot {\pi}^{2} + {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{2}\right) + \frac{1}{2} \cdot \left(\pi \cdot \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\frac{1}{8} \cdot {\pi}^{3} - {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\frac{1}{4} \cdot {\pi}^{2} + {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{2}\right) + \frac{1}{2} \cdot \left(\pi \cdot \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}
double f(double v) {
        double r267170 = 1.0;
        double r267171 = 5.0;
        double r267172 = v;
        double r267173 = r267172 * r267172;
        double r267174 = r267171 * r267173;
        double r267175 = r267170 - r267174;
        double r267176 = r267173 - r267170;
        double r267177 = r267175 / r267176;
        double r267178 = acos(r267177);
        return r267178;
}
↓
double f(double v) {
        double r267179 = 0.125;
        double r267180 = atan2(1.0, 0.0);
        double r267181 = 3.0;
        double r267182 = pow(r267180, r267181);
        double r267183 = r267179 * r267182;
        double r267184 = 4.0;
        double r267185 = v;
        double r267186 = r267185 * r267185;
        double r267187 = 4.0;
        double r267188 = pow(r267185, r267187);
        double r267189 = r267186 + r267188;
        double r267190 = r267184 * r267189;
        double r267191 = 1.0;
        double r267192 = r267190 - r267191;
        double r267193 = asin(r267192);
        double r267194 = pow(r267193, r267181);
        double r267195 = r267183 - r267194;
        double r267196 = 0.25;
        double r267197 = 2.0;
        double r267198 = pow(r267180, r267197);
        double r267199 = r267196 * r267198;
        double r267200 = pow(r267193, r267197);
        double r267201 = r267199 + r267200;
        double r267202 = 0.5;
        double r267203 = r267180 * r267193;
        double r267204 = r267202 * r267203;
        double r267205 = r267201 + r267204;
        double r267206 = r267195 / r267205;
        return r267206;
}

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. Taylor expanded around 0 0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\]

  3. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)}\]
  4. Using strategy rm

  5. Applied acos-asin0.8

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

  7. Applied flip3--0.8

    \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right) \cdot \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}}\]

  8. Simplified0.8

    \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{3}}{\color{blue}{\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right) + \frac{\pi}{4} \cdot \pi}}\]

  9. Taylor expanded around 0 0.8

    \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot {\pi}^{3} - {\left(\sin^{-1} \left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)\right)}^{3}}{\frac{1}{4} \cdot {\pi}^{2} + \left({\left(\sin^{-1} \left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)\right)}^{2} + \frac{1}{2} \cdot \left(\sin^{-1} \left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right) \cdot \pi\right)\right)}}\]

  10. Simplified0.8

    \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot {\pi}^{3} - {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\frac{1}{4} \cdot {\pi}^{2} + {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{2}\right) + \frac{1}{2} \cdot \left(\pi \cdot \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}}\]

  11. Final simplification0.8

    \[\leadsto \frac{\frac{1}{8} \cdot {\pi}^{3} - {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{3}}{\left(\frac{1}{4} \cdot {\pi}^{2} + {\left(\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}^{2}\right) + \frac{1}{2} \cdot \left(\pi \cdot \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\right)}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))