Average Error: 7.4 → 0.9
Time: 4.1s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.86218541740101406 \cdot 10^{182} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.83663500979720747 \cdot 10^{215}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.86218541740101406 \cdot 10^{182} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.83663500979720747 \cdot 10^{215}\right):\\
\;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r649452 = x;
        double r649453 = y;
        double r649454 = r649452 * r649453;
        double r649455 = z;
        double r649456 = 9.0;
        double r649457 = r649455 * r649456;
        double r649458 = t;
        double r649459 = r649457 * r649458;
        double r649460 = r649454 - r649459;
        double r649461 = a;
        double r649462 = 2.0;
        double r649463 = r649461 * r649462;
        double r649464 = r649460 / r649463;
        return r649464;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r649465 = x;
        double r649466 = y;
        double r649467 = r649465 * r649466;
        double r649468 = z;
        double r649469 = 9.0;
        double r649470 = r649468 * r649469;
        double r649471 = t;
        double r649472 = r649470 * r649471;
        double r649473 = r649467 - r649472;
        double r649474 = -1.862185417401014e+182;
        bool r649475 = r649473 <= r649474;
        double r649476 = 7.836635009797207e+215;
        bool r649477 = r649473 <= r649476;
        double r649478 = !r649477;
        bool r649479 = r649475 || r649478;
        double r649480 = 0.5;
        double r649481 = r649465 * r649480;
        double r649482 = a;
        double r649483 = r649466 / r649482;
        double r649484 = r649481 * r649483;
        double r649485 = 4.5;
        double r649486 = r649471 * r649485;
        double r649487 = r649468 / r649482;
        double r649488 = r649486 * r649487;
        double r649489 = r649484 - r649488;
        double r649490 = r649467 / r649482;
        double r649491 = r649480 * r649490;
        double r649492 = r649471 * r649468;
        double r649493 = r649492 / r649482;
        double r649494 = r649485 * r649493;
        double r649495 = r649491 - r649494;
        double r649496 = r649479 ? r649489 : r649495;
        return r649496;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.4
Target5.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -1.862185417401014e+182 or 7.836635009797207e+215 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 27.4

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    2. Taylor expanded around 0 26.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity26.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

    5. Applied times-frac14.5

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

    6. Applied associate-*r*14.5

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

    7. Simplified14.5

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(t \cdot 4.5\right)} \cdot \frac{z}{a}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity14.5

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    10. Applied times-frac1.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    11. Applied associate-*r*1.4

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{1}\right) \cdot \frac{y}{a}} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    12. Simplified1.4

      \[\leadsto \color{blue}{\left(x \cdot 0.5\right)} \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    if -1.862185417401014e+182 < (- (* x y) (* (* z 9.0) t)) < 7.836635009797207e+215

    1. Initial program 0.7

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    2. Taylor expanded around 0 0.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity0.7

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

    5. Applied times-frac5.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

    6. Applied associate-*r*5.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

    7. Simplified5.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(t \cdot 4.5\right)} \cdot \frac{z}{a}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity5.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{\color{blue}{1 \cdot a}}\]

    10. Applied add-cube-cbrt6.3

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{1 \cdot a}\]

    11. Applied times-frac6.3

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{a}\right)}\]

    12. Applied associate-*r*2.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(\left(t \cdot 4.5\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1}\right) \cdot \frac{\sqrt[3]{z}}{a}}\]

    13. Simplified2.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(t \cdot \left(4.5 \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)} \cdot \frac{\sqrt[3]{z}}{a}\]

    14. Taylor expanded around 0 0.7

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{4.5 \cdot \frac{t \cdot z}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.86218541740101406 \cdot 10^{182} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 7.83663500979720747 \cdot 10^{215}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))