Average Error: 0.3 → 0.3
Time: 8.0s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right)}^{t}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)\]
\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(\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right)}^{t}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)
double f(double x, double y, double z, double t) {
        double r707615 = x;
        double r707616 = 0.5;
        double r707617 = r707615 * r707616;
        double r707618 = y;
        double r707619 = r707617 - r707618;
        double r707620 = z;
        double r707621 = 2.0;
        double r707622 = r707620 * r707621;
        double r707623 = sqrt(r707622);
        double r707624 = r707619 * r707623;
        double r707625 = t;
        double r707626 = r707625 * r707625;
        double r707627 = r707626 / r707621;
        double r707628 = exp(r707627);
        double r707629 = r707624 * r707628;
        return r707629;
}
↓
double f(double x, double y, double z, double t) {
        double r707630 = x;
        double r707631 = 0.5;
        double r707632 = r707630 * r707631;
        double r707633 = y;
        double r707634 = r707632 - r707633;
        double r707635 = z;
        double r707636 = 2.0;
        double r707637 = r707635 * r707636;
        double r707638 = sqrt(r707637);
        double r707639 = r707634 * r707638;
        double r707640 = t;
        double r707641 = exp(r707640);
        double r707642 = pow(r707641, r707640);
        double r707643 = cbrt(r707642);
        double r707644 = cbrt(r707641);
        double r707645 = r707644 * r707644;
        double r707646 = pow(r707645, r707640);
        double r707647 = cbrt(r707646);
        double r707648 = r707643 * r707647;
        double r707649 = pow(r707644, r707640);
        double r707650 = cbrt(r707649);
        double r707651 = r707648 * r707650;
        double r707652 = 1.0;
        double r707653 = r707652 / r707636;
        double r707654 = pow(r707651, r707653);
        double r707655 = pow(r707643, r707653);
        double r707656 = r707654 * r707655;
        double r707657 = r707639 * r707656;
        return r707657;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Target

Original0.3
Target0.3
Herbie0.3
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)}\]

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-identity0.3

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

  4. Applied times-frac0.3

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

  5. Applied exp-prod0.3

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

  6. Simplified0.3

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

  8. Applied div-inv0.3

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

  9. Applied pow-unpow0.3

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

  11. Applied add-cube-cbrt0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\left(e^{t}\right)}^{t}}\right) \cdot \sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}}^{\left(\frac{1}{2}\right)}\]

  12. Applied unpow-prod-down0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)}\]
  13. Using strategy rm

  14. Applied add-cube-cbrt0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right) \cdot \sqrt[3]{e^{t}}\right)}}^{t}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)\]

  15. Applied unpow-prod-down0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right)}^{t} \cdot {\left(\sqrt[3]{e^{t}}\right)}^{t}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)\]

  16. Applied cbrt-prod0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right)}^{t}} \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}}\right)}^{t}}\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)\]

  17. Applied associate-*r*0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\color{blue}{\left(\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right)}^{t}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}}\right)}^{t}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)\]

  18. Final simplification0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\left(\sqrt[3]{{\left(e^{t}\right)}^{t}} \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}} \cdot \sqrt[3]{e^{t}}\right)}^{t}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{e^{t}}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{{\left(e^{t}\right)}^{t}}\right)}^{\left(\frac{1}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020020 +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))))