Average Error: 3.9 → 1.8
Time: 21.8s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\frac{\frac{\frac{t}{z}}{3}}{y} + \left(\left(\frac{\frac{y}{3}}{z} + y \cdot \frac{\frac{-1}{3}}{z}\right) + \left(x + \frac{\frac{-y}{3}}{z}\right)\right)\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\frac{\frac{\frac{t}{z}}{3}}{y} + \left(\left(\frac{\frac{y}{3}}{z} + y \cdot \frac{\frac{-1}{3}}{z}\right) + \left(x + \frac{\frac{-y}{3}}{z}\right)\right)
double f(double x, double y, double z, double t) {
        double r20898565 = x;
        double r20898566 = y;
        double r20898567 = z;
        double r20898568 = 3.0;
        double r20898569 = r20898567 * r20898568;
        double r20898570 = r20898566 / r20898569;
        double r20898571 = r20898565 - r20898570;
        double r20898572 = t;
        double r20898573 = r20898569 * r20898566;
        double r20898574 = r20898572 / r20898573;
        double r20898575 = r20898571 + r20898574;
        return r20898575;
}
↓
double f(double x, double y, double z, double t) {
        double r20898576 = t;
        double r20898577 = z;
        double r20898578 = r20898576 / r20898577;
        double r20898579 = 3.0;
        double r20898580 = r20898578 / r20898579;
        double r20898581 = y;
        double r20898582 = r20898580 / r20898581;
        double r20898583 = r20898581 / r20898579;
        double r20898584 = r20898583 / r20898577;
        double r20898585 = -1.0;
        double r20898586 = r20898585 / r20898579;
        double r20898587 = r20898586 / r20898577;
        double r20898588 = r20898581 * r20898587;
        double r20898589 = r20898584 + r20898588;
        double r20898590 = x;
        double r20898591 = -r20898581;
        double r20898592 = r20898591 / r20898579;
        double r20898593 = r20898592 / r20898577;
        double r20898594 = r20898590 + r20898593;
        double r20898595 = r20898589 + r20898594;
        double r20898596 = r20898582 + r20898595;
        return r20898596;
}

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

Original3.9
Target1.7
Herbie1.8
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Initial program 3.9

    \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
  2. Using strategy rm

  3. Applied associate-/r*1.7

    \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}}\]
  4. Using strategy rm

  5. Applied associate-/r*1.7

    \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{\frac{t}{z}}{3}}}{y}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity1.7

    \[\leadsto \left(x - \frac{\color{blue}{1 \cdot y}}{z \cdot 3}\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  8. Applied times-frac1.7

    \[\leadsto \left(x - \color{blue}{\frac{1}{z} \cdot \frac{y}{3}}\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  9. Applied add-sqr-sqrt32.5

    \[\leadsto \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  10. Applied prod-diff32.5

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\frac{y}{3} \cdot \frac{1}{z}\right) + \mathsf{fma}\left(-\frac{y}{3}, \frac{1}{z}, \frac{y}{3} \cdot \frac{1}{z}\right)\right)} + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  11. Simplified1.7

    \[\leadsto \left(\color{blue}{\left(x + \left(-\frac{\frac{y}{3}}{z}\right)\right)} + \mathsf{fma}\left(-\frac{y}{3}, \frac{1}{z}, \frac{y}{3} \cdot \frac{1}{z}\right)\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  12. Simplified1.7

    \[\leadsto \left(\left(x + \left(-\frac{\frac{y}{3}}{z}\right)\right) + \color{blue}{\left(\left(-\frac{\frac{y}{3}}{z}\right) + \frac{\frac{y}{3}}{z}\right)}\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]
  13. Using strategy rm

  14. Applied *-un-lft-identity1.7

    \[\leadsto \left(\left(x + \left(-\frac{\frac{y}{3}}{z}\right)\right) + \left(\left(-\frac{\frac{y}{3}}{\color{blue}{1 \cdot z}}\right) + \frac{\frac{y}{3}}{z}\right)\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  15. Applied div-inv1.7

    \[\leadsto \left(\left(x + \left(-\frac{\frac{y}{3}}{z}\right)\right) + \left(\left(-\frac{\color{blue}{y \cdot \frac{1}{3}}}{1 \cdot z}\right) + \frac{\frac{y}{3}}{z}\right)\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  16. Applied times-frac1.8

    \[\leadsto \left(\left(x + \left(-\frac{\frac{y}{3}}{z}\right)\right) + \left(\left(-\color{blue}{\frac{y}{1} \cdot \frac{\frac{1}{3}}{z}}\right) + \frac{\frac{y}{3}}{z}\right)\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  17. Simplified1.8

    \[\leadsto \left(\left(x + \left(-\frac{\frac{y}{3}}{z}\right)\right) + \left(\left(-\color{blue}{y} \cdot \frac{\frac{1}{3}}{z}\right) + \frac{\frac{y}{3}}{z}\right)\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  18. Final simplification1.8

    \[\leadsto \frac{\frac{\frac{t}{z}}{3}}{y} + \left(\left(\frac{\frac{y}{3}}{z} + y \cdot \frac{\frac{-1}{3}}{z}\right) + \left(x + \frac{\frac{-y}{3}}{z}\right)\right)\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))