Average Error: 0.4 → 0.3
Time: 32.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
↓
\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r92202 = 1.0;
        double r92203 = 6.0;
        double r92204 = r92202 / r92203;
        double r92205 = -2.0;
        double r92206 = u1;
        double r92207 = log(r92206);
        double r92208 = r92205 * r92207;
        double r92209 = 0.5;
        double r92210 = pow(r92208, r92209);
        double r92211 = r92204 * r92210;
        double r92212 = 2.0;
        double r92213 = atan2(1.0, 0.0);
        double r92214 = r92212 * r92213;
        double r92215 = u2;
        double r92216 = r92214 * r92215;
        double r92217 = cos(r92216);
        double r92218 = r92211 * r92217;
        double r92219 = r92218 + r92209;
        return r92219;
}
↓
double f(double u1, double u2) {
        double r92220 = 1.0;
        double r92221 = 6.0;
        double r92222 = r92220 / r92221;
        double r92223 = sqrt(r92222);
        double r92224 = -2.0;
        double r92225 = u1;
        double r92226 = log(r92225);
        double r92227 = r92224 * r92226;
        double r92228 = 0.5;
        double r92229 = pow(r92227, r92228);
        double r92230 = r92223 * r92229;
        double r92231 = r92223 * r92230;
        double r92232 = 2.0;
        double r92233 = atan2(1.0, 0.0);
        double r92234 = r92232 * r92233;
        double r92235 = u2;
        double r92236 = r92234 * r92235;
        double r92237 = cos(r92236);
        double r92238 = r92231 * r92237;
        double r92239 = r92238 + r92228;
        return r92239;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.4

    \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{6}} \cdot \sqrt{\frac{1}{6}}\right)} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  4. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  5. Final simplification0.3

    \[\leadsto \left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019208 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))