Average Error: 7.5 → 4.0
Time: 3.6s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -4.10488348383086384 \cdot 10^{165}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.4653069814584246 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.90384403871874761 \cdot 10^{297}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -4.10488348383086384 \cdot 10^{165}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.4653069814584246 \cdot 10^{-58}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{4.5}{a} \cdot \left(t \cdot z\right)\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.90384403871874761 \cdot 10^{297}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r542783 = x;
        double r542784 = y;
        double r542785 = r542783 * r542784;
        double r542786 = z;
        double r542787 = 9.0;
        double r542788 = r542786 * r542787;
        double r542789 = t;
        double r542790 = r542788 * r542789;
        double r542791 = r542785 - r542790;
        double r542792 = a;
        double r542793 = 2.0;
        double r542794 = r542792 * r542793;
        double r542795 = r542791 / r542794;
        return r542795;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r542796 = z;
        double r542797 = 9.0;
        double r542798 = r542796 * r542797;
        double r542799 = t;
        double r542800 = r542798 * r542799;
        double r542801 = -inf.0;
        bool r542802 = r542800 <= r542801;
        double r542803 = 0.5;
        double r542804 = x;
        double r542805 = y;
        double r542806 = r542804 * r542805;
        double r542807 = a;
        double r542808 = r542806 / r542807;
        double r542809 = r542803 * r542808;
        double r542810 = 4.5;
        double r542811 = r542799 * r542810;
        double r542812 = r542796 / r542807;
        double r542813 = r542811 * r542812;
        double r542814 = r542809 - r542813;
        double r542815 = -4.104883483830864e+165;
        bool r542816 = r542800 <= r542815;
        double r542817 = r542805 / r542807;
        double r542818 = r542804 * r542817;
        double r542819 = r542803 * r542818;
        double r542820 = r542799 * r542796;
        double r542821 = r542820 / r542807;
        double r542822 = r542810 * r542821;
        double r542823 = r542819 - r542822;
        double r542824 = 1.4653069814584246e-58;
        bool r542825 = r542800 <= r542824;
        double r542826 = r542810 / r542807;
        double r542827 = r542826 * r542820;
        double r542828 = r542809 - r542827;
        double r542829 = 1.9038440387187476e+297;
        bool r542830 = r542800 <= r542829;
        double r542831 = r542830 ? r542823 : r542814;
        double r542832 = r542825 ? r542828 : r542831;
        double r542833 = r542816 ? r542823 : r542832;
        double r542834 = r542802 ? r542814 : r542833;
        return r542834;
}

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

Original7.5
Target5.7
Herbie4.0
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* z 9.0) t) < -inf.0 or 1.9038440387187476e+297 < (* (* z 9.0) t)

    1. Initial program 59.8

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

    2. Taylor expanded around 0 58.9

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

    4. Applied *-un-lft-identity58.9

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

    5. Applied times-frac7.0

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

    6. Applied associate-*r*7.4

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

    7. Simplified7.4

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

    if -inf.0 < (* (* z 9.0) t) < -4.104883483830864e+165 or 1.4653069814584246e-58 < (* (* z 9.0) t) < 1.9038440387187476e+297

    1. Initial program 4.5

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

    2. Taylor expanded around 0 4.5

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

    4. Applied *-un-lft-identity4.5

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

    5. Applied times-frac2.9

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

    6. Simplified2.9

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

    if -4.104883483830864e+165 < (* (* z 9.0) t) < 1.4653069814584246e-58

    1. Initial program 4.2

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

    2. Taylor expanded around 0 4.2

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

    4. Applied clear-num4.4

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\frac{1}{\frac{a}{t \cdot z}}}\]
    5. Using strategy rm

    6. Applied associate-/r/4.2

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

    7. Applied associate-*r*4.2

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

    8. Simplified4.2

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -4.10488348383086384 \cdot 10^{165}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.4653069814584246 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.90384403871874761 \cdot 10^{297}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))