Average Error: 7.9 → 0.5
Time: 4.3s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.570465535572884322699978790782884605691 \cdot 10^{-101} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.47440313991068531415406912842436761104 \cdot 10^{-267} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.212678920874577831195231744431104229608 \cdot 10^{275}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\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 = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.570465535572884322699978790782884605691 \cdot 10^{-101} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.47440313991068531415406912842436761104 \cdot 10^{-267} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.212678920874577831195231744431104229608 \cdot 10^{275}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\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 r778392 = x;
        double r778393 = y;
        double r778394 = r778392 * r778393;
        double r778395 = z;
        double r778396 = 9.0;
        double r778397 = r778395 * r778396;
        double r778398 = t;
        double r778399 = r778397 * r778398;
        double r778400 = r778394 - r778399;
        double r778401 = a;
        double r778402 = 2.0;
        double r778403 = r778401 * r778402;
        double r778404 = r778400 / r778403;
        return r778404;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r778405 = x;
        double r778406 = y;
        double r778407 = r778405 * r778406;
        double r778408 = z;
        double r778409 = 9.0;
        double r778410 = r778408 * r778409;
        double r778411 = t;
        double r778412 = r778410 * r778411;
        double r778413 = r778407 - r778412;
        double r778414 = -inf.0;
        bool r778415 = r778413 <= r778414;
        double r778416 = -1.5704655355728843e-101;
        bool r778417 = r778413 <= r778416;
        double r778418 = 1.4744031399106853e-267;
        bool r778419 = r778413 <= r778418;
        double r778420 = 6.212678920874578e+275;
        bool r778421 = r778413 <= r778420;
        double r778422 = !r778421;
        bool r778423 = r778419 || r778422;
        double r778424 = !r778423;
        bool r778425 = r778417 || r778424;
        double r778426 = !r778425;
        bool r778427 = r778415 || r778426;
        double r778428 = 0.5;
        double r778429 = a;
        double r778430 = r778429 / r778406;
        double r778431 = r778405 / r778430;
        double r778432 = r778428 * r778431;
        double r778433 = 4.5;
        double r778434 = r778433 * r778411;
        double r778435 = r778408 / r778429;
        double r778436 = r778434 * r778435;
        double r778437 = r778432 - r778436;
        double r778438 = r778407 / r778429;
        double r778439 = r778428 * r778438;
        double r778440 = r778411 * r778408;
        double r778441 = r778440 / r778429;
        double r778442 = r778433 * r778441;
        double r778443 = r778439 - r778442;
        double r778444 = r778427 ? r778437 : r778443;
        return r778444;
}

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

Original7.9
Target5.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or -1.5704655355728843e-101 < (- (* x y) (* (* z 9.0) t)) < 1.4744031399106853e-267 or 6.212678920874578e+275 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 33.8

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

    2. Taylor expanded around 0 33.5

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

    4. Applied *-un-lft-identity33.5

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

    5. Applied times-frac19.0

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

    6. Simplified19.0

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

    8. Applied associate-*r*19.1

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

    10. Applied associate-/l*1.3

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

    if -inf.0 < (- (* x y) (* (* z 9.0) t)) < -1.5704655355728843e-101 or 1.4744031399106853e-267 < (- (* x y) (* (* z 9.0) t)) < 6.212678920874578e+275

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.3

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

    4. Applied *-un-lft-identity0.3

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

    5. Applied times-frac5.4

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

    6. Simplified5.4

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

    8. Applied associate-*r*5.4

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

    9. Taylor expanded around 0 0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.570465535572884322699978790782884605691 \cdot 10^{-101} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.47440313991068531415406912842436761104 \cdot 10^{-267} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.212678920874577831195231744431104229608 \cdot 10^{275}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))