Average Error: 0.3 → 0.5
Time: 22.5s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot e^{\frac{t \cdot t}{2}}\]
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
↓
\left(\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot e^{\frac{t \cdot t}{2}}
double f(double x, double y, double z, double t) {
        double r526938 = x;
        double r526939 = 0.5;
        double r526940 = r526938 * r526939;
        double r526941 = y;
        double r526942 = r526940 - r526941;
        double r526943 = z;
        double r526944 = 2.0;
        double r526945 = r526943 * r526944;
        double r526946 = sqrt(r526945);
        double r526947 = r526942 * r526946;
        double r526948 = t;
        double r526949 = r526948 * r526948;
        double r526950 = r526949 / r526944;
        double r526951 = exp(r526950);
        double r526952 = r526947 * r526951;
        return r526952;
}
↓
double f(double x, double y, double z, double t) {
        double r526953 = x;
        double r526954 = 0.5;
        double r526955 = r526953 * r526954;
        double r526956 = y;
        double r526957 = r526955 - r526956;
        double r526958 = z;
        double r526959 = sqrt(r526958);
        double r526960 = r526957 * r526959;
        double r526961 = 2.0;
        double r526962 = sqrt(r526961);
        double r526963 = sqrt(r526962);
        double r526964 = r526960 * r526963;
        double r526965 = r526964 * r526963;
        double r526966 = t;
        double r526967 = r526966 * r526966;
        double r526968 = r526967 / r526961;
        double r526969 = exp(r526968);
        double r526970 = r526965 * r526969;
        return r526970;
}

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 add-sqr-sqrt0.5

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

  7. Applied sqrt-prod0.6

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

  8. Applied associate-*r*0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019347 +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))))