Average Error: 7.9 → 0.5
Time: 4.2s
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 r653539 = x;
        double r653540 = y;
        double r653541 = r653539 * r653540;
        double r653542 = z;
        double r653543 = 9.0;
        double r653544 = r653542 * r653543;
        double r653545 = t;
        double r653546 = r653544 * r653545;
        double r653547 = r653541 - r653546;
        double r653548 = a;
        double r653549 = 2.0;
        double r653550 = r653548 * r653549;
        double r653551 = r653547 / r653550;
        return r653551;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r653552 = x;
        double r653553 = y;
        double r653554 = r653552 * r653553;
        double r653555 = z;
        double r653556 = 9.0;
        double r653557 = r653555 * r653556;
        double r653558 = t;
        double r653559 = r653557 * r653558;
        double r653560 = r653554 - r653559;
        double r653561 = -inf.0;
        bool r653562 = r653560 <= r653561;
        double r653563 = -1.5704655355728843e-101;
        bool r653564 = r653560 <= r653563;
        double r653565 = 1.4744031399106853e-267;
        bool r653566 = r653560 <= r653565;
        double r653567 = 6.212678920874578e+275;
        bool r653568 = r653560 <= r653567;
        double r653569 = !r653568;
        bool r653570 = r653566 || r653569;
        double r653571 = !r653570;
        bool r653572 = r653564 || r653571;
        double r653573 = !r653572;
        bool r653574 = r653562 || r653573;
        double r653575 = 0.5;
        double r653576 = a;
        double r653577 = r653576 / r653553;
        double r653578 = r653552 / r653577;
        double r653579 = r653575 * r653578;
        double r653580 = 4.5;
        double r653581 = r653580 * r653558;
        double r653582 = r653555 / r653576;
        double r653583 = r653581 * r653582;
        double r653584 = r653579 - r653583;
        double r653585 = r653554 / r653576;
        double r653586 = r653575 * r653585;
        double r653587 = r653558 * r653555;
        double r653588 = r653587 / r653576;
        double r653589 = r653580 * r653588;
        double r653590 = r653586 - r653589;
        double r653591 = r653574 ? r653584 : r653590;
        return r653591;
}

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 +o rules:numerics
(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)))