Average Error: 0.3 → 0.5
Time: 15.3s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}\right) \cdot e^{\frac{\frac{t \cdot t}{\sqrt{2}}}{\sqrt{2}}}\]
\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^{\frac{\frac{t \cdot t}{\sqrt{2}}}{\sqrt{2}}}
double f(double x, double y, double z, double t) {
        double r884564 = x;
        double r884565 = 0.5;
        double r884566 = r884564 * r884565;
        double r884567 = y;
        double r884568 = r884566 - r884567;
        double r884569 = z;
        double r884570 = 2.0;
        double r884571 = r884569 * r884570;
        double r884572 = sqrt(r884571);
        double r884573 = r884568 * r884572;
        double r884574 = t;
        double r884575 = r884574 * r884574;
        double r884576 = r884575 / r884570;
        double r884577 = exp(r884576);
        double r884578 = r884573 * r884577;
        return r884578;
}
↓
double f(double x, double y, double z, double t) {
        double r884579 = x;
        double r884580 = 0.5;
        double r884581 = r884579 * r884580;
        double r884582 = y;
        double r884583 = r884581 - r884582;
        double r884584 = z;
        double r884585 = sqrt(r884584);
        double r884586 = r884583 * r884585;
        double r884587 = 2.0;
        double r884588 = sqrt(r884587);
        double r884589 = r884586 * r884588;
        double r884590 = t;
        double r884591 = r884590 * r884590;
        double r884592 = r884591 / r884588;
        double r884593 = r884592 / r884588;
        double r884594 = exp(r884593);
        double r884595 = r884589 * r884594;
        return r884595;
}

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

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

  7. Applied associate-/r*0.5

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

  8. Final simplification0.5

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

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))))