Average Error: 0.5 → 0.5
Time: 4.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}\right)} - 1\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}\right)} - 1
double f(double v) {
        double r262202 = 1.0;
        double r262203 = 5.0;
        double r262204 = v;
        double r262205 = r262204 * r262204;
        double r262206 = r262203 * r262205;
        double r262207 = r262202 - r262206;
        double r262208 = r262205 - r262202;
        double r262209 = r262207 / r262208;
        double r262210 = acos(r262209);
        return r262210;
}
↓
double f(double v) {
        double r262211 = 1.0;
        double r262212 = 5.0;
        double r262213 = v;
        double r262214 = 2.0;
        double r262215 = pow(r262213, r262214);
        double r262216 = r262212 * r262215;
        double r262217 = r262211 - r262216;
        double r262218 = r262215 - r262211;
        double r262219 = r262217 / r262218;
        double r262220 = acos(r262219);
        double r262221 = 3.0;
        double r262222 = pow(r262220, r262221);
        double r262223 = cbrt(r262222);
        double r262224 = log1p(r262223);
        double r262225 = exp(r262224);
        double r262226 = 1.0;
        double r262227 = r262225 - r262226;
        return r262227;
}

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 expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied expm1-udef0.5

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} - 1}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube0.5

    \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\right)} - 1\]

  8. Simplified0.5

    \[\leadsto e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}}\right)} - 1\]

  9. Final simplification0.5

    \[\leadsto e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}\right)} - 1\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))