Average Error: 0.3 → 0.5
Time: 8.5s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\left(\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot e^{\frac{t \cdot t}{2}}\]
\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(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot e^{\frac{t \cdot t}{2}}
double f(double x, double y, double z, double t) {
        double r789581 = x;
        double r789582 = 0.5;
        double r789583 = r789581 * r789582;
        double r789584 = y;
        double r789585 = r789583 - r789584;
        double r789586 = z;
        double r789587 = 2.0;
        double r789588 = r789586 * r789587;
        double r789589 = sqrt(r789588);
        double r789590 = r789585 * r789589;
        double r789591 = t;
        double r789592 = r789591 * r789591;
        double r789593 = r789592 / r789587;
        double r789594 = exp(r789593);
        double r789595 = r789590 * r789594;
        return r789595;
}
↓
double f(double x, double y, double z, double t) {
        double r789596 = x;
        double r789597 = 0.5;
        double r789598 = r789596 * r789597;
        double r789599 = y;
        double r789600 = r789598 - r789599;
        double r789601 = z;
        double r789602 = sqrt(r789601);
        double r789603 = r789600 * r789602;
        double r789604 = 2.0;
        double r789605 = sqrt(r789604);
        double r789606 = sqrt(r789605);
        double r789607 = r789603 * r789606;
        double r789608 = sqrt(r789606);
        double r789609 = r789607 * r789608;
        double r789610 = r789609 * r789608;
        double r789611 = t;
        double r789612 = r789611 * r789611;
        double r789613 = r789612 / r789604;
        double r789614 = exp(r789613);
        double r789615 = r789610 * r789614;
        return r789615;
}

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*0.6

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

  10. Applied add-sqr-sqrt0.6

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

  11. Applied sqrt-prod0.6

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

  12. Applied sqrt-prod0.6

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

  13. Applied associate-*r*0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020002 
(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))))