Average Error: 0.5 → 0.5
Time: 31.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}\right)
double f(double v) {
        double r117994 = 1.0;
        double r117995 = 5.0;
        double r117996 = v;
        double r117997 = r117996 * r117996;
        double r117998 = r117995 * r117997;
        double r117999 = r117994 - r117998;
        double r118000 = r117997 - r117994;
        double r118001 = r117999 / r118000;
        double r118002 = acos(r118001);
        return r118002;
}
↓
double f(double v) {
        double r118003 = 1.0;
        double r118004 = 5.0;
        double r118005 = v;
        double r118006 = r118005 * r118005;
        double r118007 = r118004 * r118006;
        double r118008 = r118003 - r118007;
        double r118009 = r118006 - r118003;
        double r118010 = r118008 / r118009;
        double r118011 = acos(r118010);
        double r118012 = log1p(r118011);
        double r118013 = sqrt(r118012);
        double r118014 = exp(r118007);
        double r118015 = log(r118014);
        double r118016 = r118003 - r118015;
        double r118017 = r118016 / r118009;
        double r118018 = acos(r118017);
        double r118019 = log1p(r118018);
        double r118020 = sqrt(r118019);
        double r118021 = r118013 * r118020;
        double r118022 = expm1(r118021);
        return r118022;
}

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 add-sqr-sqrt0.5

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

  7. Applied add-log-exp0.5

    \[\leadsto \mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \color{blue}{\log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}}{v \cdot v - 1}\right)\right)}\right)\]

  8. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))