Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Average Error: 0.3 → 0.3

Time: 7.8s

Precision: 64

Math

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]

↓

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)} \cdot {\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)\]

C

double f(double x, double y, double z, double t) {
        double r906149 = x;
        double r906150 = 0.5;
        double r906151 = r906149 * r906150;
        double r906152 = y;
        double r906153 = r906151 - r906152;
        double r906154 = z;
        double r906155 = 2.0;
        double r906156 = r906154 * r906155;
        double r906157 = sqrt(r906156);
        double r906158 = r906153 * r906157;
        double r906159 = t;
        double r906160 = r906159 * r906159;
        double r906161 = r906160 / r906155;
        double r906162 = exp(r906161);
        double r906163 = r906158 * r906162;
        return r906163;
}

↓

double f(double x, double y, double z, double t) {
        double r906164 = x;
        double r906165 = 0.5;
        double r906166 = r906164 * r906165;
        double r906167 = y;
        double r906168 = r906166 - r906167;
        double r906169 = z;
        double r906170 = 2.0;
        double r906171 = r906169 * r906170;
        double r906172 = sqrt(r906171);
        double r906173 = r906168 * r906172;
        double r906174 = t;
        double r906175 = exp(r906174);
        double r906176 = r906174 / r906170;
        double r906177 = 2.0;
        double r906178 = r906176 / r906177;
        double r906179 = pow(r906175, r906178);
        double r906180 = r906179 * r906179;
        double r906181 = r906173 * r906180;
        return r906181;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	0.3
Target	0.3
Herbie	0.3

\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)}\]

Derivation

1. Initial program 0.3

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
2. Using strategy rm

3. Applied *-un-lft-identity 0.3

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{\color{blue}{1 \cdot 2}}}\]

4. Applied times-frac 0.3

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t}{1} \cdot \frac{t}{2}}}\]

5. Applied exp-prod 0.3

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{\frac{t}{1}}\right)}^{\left(\frac{t}{2}\right)}}\]

6. Simplified 0.3

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\color{blue}{\left(e^{t}\right)}}^{\left(\frac{t}{2}\right)}\]
7. Using strategy rm

8. Applied add-cbrt-cube 0.3

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt[3]{\left({\left(e^{t}\right)}^{\left(\frac{t}{2}\right)} \cdot {\left(e^{t}\right)}^{\left(\frac{t}{2}\right)}\right) \cdot {\left(e^{t}\right)}^{\left(\frac{t}{2}\right)}}}\]

9. Simplified 0.3

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt[3]{\color{blue}{{\left({\left(e^{t}\right)}^{\left(\frac{t}{2}\right)}\right)}^{3}}}\]
10. Using strategy rm

11. Applied sqr-pow 0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt[3]{{\color{blue}{\left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)} \cdot {\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)}}^{3}}\]

12. Applied unpow-prod-down 0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt[3]{\color{blue}{{\left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)}^{3} \cdot {\left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)}^{3}}}\]

13. Applied cbrt-prod 0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(\sqrt[3]{{\left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)}^{3}} \cdot \sqrt[3]{{\left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)}^{3}}\right)}\]

14. Simplified 0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{{\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}} \cdot \sqrt[3]{{\left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)}^{3}}\right)\]

15. Simplified 0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)} \cdot \color{blue}{{\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}}\right)\]

16. Final simplification 0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)} \cdot {\left(e^{t}\right)}^{\left(\frac{\frac{t}{2}}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2))) (exp (/ (* t t) 2))))