Random Jason Timeout Test 012

Average Error: 61.2 → 59.0

Time: 46.6s

Precision: 64

• could not determine a ground truth for program body

  1. a = 3.9540437202843566e+32

Math

\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le 1.5597915497031036 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log \left(e^{\mathsf{log1p}\left(a\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \end{array}\]

C

double f(double a) {
        double r16170 = a;
        double r16171 = cosh(r16170);
        double r16172 = r16170 * r16170;
        double r16173 = fmod(r16171, r16172);
        double r16174 = log1p(r16170);
        double r16175 = pow(r16173, r16174);
        double r16176 = acos(r16175);
        return r16176;
}

↓

double f(double a) {
        double r16177 = a;
        double r16178 = 1.5597915497031036e-162;
        bool r16179 = r16177 <= r16178;
        double r16180 = cosh(r16177);
        double r16181 = r16177 * r16177;
        double r16182 = fmod(r16180, r16181);
        double r16183 = log1p(r16177);
        double r16184 = exp(r16183);
        double r16185 = log(r16184);
        double r16186 = pow(r16182, r16185);
        double r16187 = acos(r16186);
        double r16188 = exp(r16182);
        double r16189 = sqrt(r16188);
        double r16190 = log(r16189);
        double r16191 = r16190 + r16190;
        double r16192 = pow(r16191, r16183);
        double r16193 = acos(r16192);
        double r16194 = r16179 ? r16187 : r16193;
        return r16194;
}

Error

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if a < 1.5597915497031036e-162

   1. Initial program 64.0

      \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
   2. Using strategy rm

   3. Applied add-log-exp 61.3

      \[\leadsto \cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\color{blue}{\left(\log \left(e^{\mathsf{log1p}\left(a\right)}\right)\right)}}\right)\]

   if 1.5597915497031036e-162 < a

   1. Initial program 58.6

      \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
   2. Using strategy rm

   3. Applied add-log-exp 56.8

      \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 56.8

      \[\leadsto \cos^{-1} \left({\left(\log \color{blue}{\left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}} \cdot \sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

   6. Applied log-prod 56.8

      \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 1.5597915497031036 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log \left(e^{\mathsf{log1p}\left(a\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))