Average Error: 0.3 → 0.5
Time: 14.8s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\sqrt{2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right)\right) \cdot e^{\frac{t \cdot t}{2}}\]
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
↓
\left(\sqrt{2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right)\right) \cdot e^{\frac{t \cdot t}{2}}
double f(double x, double y, double z, double t) {
        double r892679 = x;
        double r892680 = 0.5;
        double r892681 = r892679 * r892680;
        double r892682 = y;
        double r892683 = r892681 - r892682;
        double r892684 = z;
        double r892685 = 2.0;
        double r892686 = r892684 * r892685;
        double r892687 = sqrt(r892686);
        double r892688 = r892683 * r892687;
        double r892689 = t;
        double r892690 = r892689 * r892689;
        double r892691 = r892690 / r892685;
        double r892692 = exp(r892691);
        double r892693 = r892688 * r892692;
        return r892693;
}
↓
double f(double x, double y, double z, double t) {
        double r892694 = 2.0;
        double r892695 = sqrt(r892694);
        double r892696 = x;
        double r892697 = 0.5;
        double r892698 = r892696 * r892697;
        double r892699 = y;
        double r892700 = r892698 - r892699;
        double r892701 = z;
        double r892702 = sqrt(r892701);
        double r892703 = r892700 * r892702;
        double r892704 = r892695 * r892703;
        double r892705 = t;
        double r892706 = r892705 * r892705;
        double r892707 = r892706 / r892694;
        double r892708 = exp(r892707);
        double r892709 = r892704 * r892708;
        return r892709;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.3
Target0.3
Herbie0.5
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)}\]

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 sqrt-prod0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied *-un-lft-identity0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020042 +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))))