Average Error: 34.1 → 33.7
Time: 14.8s
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[\sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
\sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}
double f(double a, double c) {
        double r16128 = c;
        double r16129 = cosh(r16128);
        double r16130 = a;
        double r16131 = log1p(r16130);
        double r16132 = fmod(r16129, r16131);
        return r16132;
}
↓
double f(double a, double c) {
        double r16133 = c;
        double r16134 = cosh(r16133);
        double r16135 = sqrt(r16134);
        double r16136 = -1.0;
        double r16137 = r16136 * r16133;
        double r16138 = exp(r16137);
        double r16139 = exp(r16133);
        double r16140 = r16138 + r16139;
        double r16141 = sqrt(r16140);
        double r16142 = 0.5;
        double r16143 = sqrt(r16142);
        double r16144 = r16141 * r16143;
        double r16145 = r16135 * r16144;
        double r16146 = a;
        double r16147 = log1p(r16146);
        double r16148 = fmod(r16145, r16147);
        double r16149 = sqrt(r16148);
        double r16150 = r16149 * r16149;
        return r16150;
}

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.1

    \[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt34.1

    \[\leadsto \left(\color{blue}{\left(\sqrt{\cosh c} \cdot \sqrt{\cosh c}\right)} \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  4. Taylor expanded around inf 34.0

    \[\leadsto \left(\left(\sqrt{\cosh c} \cdot \color{blue}{\left(\sqrt{e^{c} + e^{-c}} \cdot \sqrt{\frac{1}{2}}\right)}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  5. Simplified34.0

    \[\leadsto \left(\left(\sqrt{\cosh c} \cdot \color{blue}{\left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt33.7

    \[\leadsto \color{blue}{\sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]

  8. Final simplification33.7

    \[\leadsto \sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \sqrt{\left(\left(\sqrt{\cosh c} \cdot \left(\sqrt{e^{-1 \cdot c} + e^{c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  :precision binary64
  (fmod (cosh c) (log1p a)))