Average Error: 0.3 → 0.5
Time: 16.2s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}\right) \cdot e^{0.5 \cdot {t}^{2}}\]
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
↓
\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}\right) \cdot e^{0.5 \cdot {t}^{2}}
double f(double x, double y, double z, double t) {
        double r814233 = x;
        double r814234 = 0.5;
        double r814235 = r814233 * r814234;
        double r814236 = y;
        double r814237 = r814235 - r814236;
        double r814238 = z;
        double r814239 = 2.0;
        double r814240 = r814238 * r814239;
        double r814241 = sqrt(r814240);
        double r814242 = r814237 * r814241;
        double r814243 = t;
        double r814244 = r814243 * r814243;
        double r814245 = r814244 / r814239;
        double r814246 = exp(r814245);
        double r814247 = r814242 * r814246;
        return r814247;
}
↓
double f(double x, double y, double z, double t) {
        double r814248 = x;
        double r814249 = 0.5;
        double r814250 = r814248 * r814249;
        double r814251 = y;
        double r814252 = r814250 - r814251;
        double r814253 = z;
        double r814254 = sqrt(r814253);
        double r814255 = r814252 * r814254;
        double r814256 = 2.0;
        double r814257 = sqrt(r814256);
        double r814258 = r814255 * r814257;
        double r814259 = t;
        double r814260 = 2.0;
        double r814261 = pow(r814259, r814260);
        double r814262 = r814249 * r814261;
        double r814263 = exp(r814262);
        double r814264 = r814258 * r814263;
        return r814264;
}

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. Taylor expanded around 0 0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020042 
(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))))