Average Error: 7.8 → 4.3
Time: 3.4s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{elif}\;x \cdot y \le -5.7260618197076615 \cdot 10^{109}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \le 2.19718171022247432 \cdot 10^{260}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y = -\infty:\\
\;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\

\mathbf{elif}\;x \cdot y \le -5.7260618197076615 \cdot 10^{109}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{elif}\;x \cdot y \le 2.19718171022247432 \cdot 10^{260}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{2}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r721816 = x;
        double r721817 = y;
        double r721818 = r721816 * r721817;
        double r721819 = z;
        double r721820 = 9.0;
        double r721821 = r721819 * r721820;
        double r721822 = t;
        double r721823 = r721821 * r721822;
        double r721824 = r721818 - r721823;
        double r721825 = a;
        double r721826 = 2.0;
        double r721827 = r721825 * r721826;
        double r721828 = r721824 / r721827;
        return r721828;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r721829 = x;
        double r721830 = y;
        double r721831 = r721829 * r721830;
        double r721832 = -inf.0;
        bool r721833 = r721831 <= r721832;
        double r721834 = 0.5;
        double r721835 = r721829 * r721834;
        double r721836 = a;
        double r721837 = r721830 / r721836;
        double r721838 = r721835 * r721837;
        double r721839 = 4.5;
        double r721840 = t;
        double r721841 = z;
        double r721842 = r721840 * r721841;
        double r721843 = r721839 * r721842;
        double r721844 = r721843 / r721836;
        double r721845 = r721838 - r721844;
        double r721846 = -5.7260618197076615e+109;
        bool r721847 = r721831 <= r721846;
        double r721848 = r721831 / r721836;
        double r721849 = r721834 * r721848;
        double r721850 = r721836 / r721841;
        double r721851 = r721840 / r721850;
        double r721852 = r721839 * r721851;
        double r721853 = r721849 - r721852;
        double r721854 = 2.1971817102224743e+260;
        bool r721855 = r721831 <= r721854;
        double r721856 = 1.0;
        double r721857 = r721856 / r721836;
        double r721858 = 9.0;
        double r721859 = r721858 * r721840;
        double r721860 = r721841 * r721859;
        double r721861 = r721831 - r721860;
        double r721862 = 2.0;
        double r721863 = r721861 / r721862;
        double r721864 = r721857 * r721863;
        double r721865 = r721855 ? r721864 : r721845;
        double r721866 = r721847 ? r721853 : r721865;
        double r721867 = r721833 ? r721845 : r721866;
        return r721867;
}

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

Original7.8
Target5.5
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -inf.0 or 2.1971817102224743e+260 < (* x y)

    1. Initial program 50.9

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

    2. Taylor expanded around 0 50.8

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

    4. Applied associate-*r/50.9

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

    6. Applied *-un-lft-identity50.9

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

    7. Applied times-frac6.3

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

    8. Applied associate-*r*6.3

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

    9. Simplified6.3

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

    if -inf.0 < (* x y) < -5.7260618197076615e+109

    1. Initial program 5.9

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

    2. Taylor expanded around 0 5.6

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

    4. Applied associate-/l*1.3

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

    if -5.7260618197076615e+109 < (* x y) < 2.1971817102224743e+260

    1. Initial program 4.4

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

    3. Applied associate-*l*4.4

      \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2}\]
    4. Using strategy rm

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

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - z \cdot \left(9 \cdot t\right)\right)}}{a \cdot 2}\]

    6. Applied times-frac4.4

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{elif}\;x \cdot y \le -5.7260618197076615 \cdot 10^{109}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \le 2.19718171022247432 \cdot 10^{260}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))