Average Error: 7.5 → 4.7
Time: 21.5s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -4.479651987078281100945877712393720755092 \cdot 10^{284}:\\ \;\;\;\;\frac{\frac{y}{2}}{a} \cdot x - \frac{9}{\frac{\frac{a \cdot 2}{t}}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -8.458949463653347121961008011969069086603 \cdot 10^{-194}:\\ \;\;\;\;\frac{x \cdot y + \left(t \cdot z\right) \cdot \left(-9\right)}{a \cdot 2}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.536566284919313133153449110235493411241 \cdot 10^{-202}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{a}{y}} - \frac{4.5}{\frac{a}{t \cdot z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.591924904207732236217241191554461171523 \cdot 10^{162}:\\ \;\;\;\;\frac{x \cdot y + \left(t \cdot z\right) \cdot \left(-9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y}{2}}{a} - \left(\frac{\frac{\sqrt{9}}{2}}{a} \cdot t\right) \cdot \left(z \cdot \sqrt{9}\right)\\ \end{array}\]
\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 \le -4.479651987078281100945877712393720755092 \cdot 10^{284}:\\
\;\;\;\;\frac{\frac{y}{2}}{a} \cdot x - \frac{9}{\frac{\frac{a \cdot 2}{t}}{z}}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -8.458949463653347121961008011969069086603 \cdot 10^{-194}:\\
\;\;\;\;\frac{x \cdot y + \left(t \cdot z\right) \cdot \left(-9\right)}{a \cdot 2}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.536566284919313133153449110235493411241 \cdot 10^{-202}:\\
\;\;\;\;\frac{0.5 \cdot x}{\frac{a}{y}} - \frac{4.5}{\frac{a}{t \cdot z}}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.591924904207732236217241191554461171523 \cdot 10^{162}:\\
\;\;\;\;\frac{x \cdot y + \left(t \cdot z\right) \cdot \left(-9\right)}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x \cdot y}{2}}{a} - \left(\frac{\frac{\sqrt{9}}{2}}{a} \cdot t\right) \cdot \left(z \cdot \sqrt{9}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r565770 = x;
        double r565771 = y;
        double r565772 = r565770 * r565771;
        double r565773 = z;
        double r565774 = 9.0;
        double r565775 = r565773 * r565774;
        double r565776 = t;
        double r565777 = r565775 * r565776;
        double r565778 = r565772 - r565777;
        double r565779 = a;
        double r565780 = 2.0;
        double r565781 = r565779 * r565780;
        double r565782 = r565778 / r565781;
        return r565782;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r565783 = z;
        double r565784 = 9.0;
        double r565785 = r565783 * r565784;
        double r565786 = t;
        double r565787 = r565785 * r565786;
        double r565788 = -4.479651987078281e+284;
        bool r565789 = r565787 <= r565788;
        double r565790 = y;
        double r565791 = 2.0;
        double r565792 = r565790 / r565791;
        double r565793 = a;
        double r565794 = r565792 / r565793;
        double r565795 = x;
        double r565796 = r565794 * r565795;
        double r565797 = r565793 * r565791;
        double r565798 = r565797 / r565786;
        double r565799 = r565798 / r565783;
        double r565800 = r565784 / r565799;
        double r565801 = r565796 - r565800;
        double r565802 = -8.458949463653347e-194;
        bool r565803 = r565787 <= r565802;
        double r565804 = r565795 * r565790;
        double r565805 = r565786 * r565783;
        double r565806 = -r565784;
        double r565807 = r565805 * r565806;
        double r565808 = r565804 + r565807;
        double r565809 = r565808 / r565797;
        double r565810 = 1.5365662849193131e-202;
        bool r565811 = r565787 <= r565810;
        double r565812 = 0.5;
        double r565813 = r565812 * r565795;
        double r565814 = r565793 / r565790;
        double r565815 = r565813 / r565814;
        double r565816 = 4.5;
        double r565817 = r565793 / r565805;
        double r565818 = r565816 / r565817;
        double r565819 = r565815 - r565818;
        double r565820 = 1.5919249042077322e+162;
        bool r565821 = r565787 <= r565820;
        double r565822 = r565804 / r565791;
        double r565823 = r565822 / r565793;
        double r565824 = sqrt(r565784);
        double r565825 = r565824 / r565791;
        double r565826 = r565825 / r565793;
        double r565827 = r565826 * r565786;
        double r565828 = r565783 * r565824;
        double r565829 = r565827 * r565828;
        double r565830 = r565823 - r565829;
        double r565831 = r565821 ? r565809 : r565830;
        double r565832 = r565811 ? r565819 : r565831;
        double r565833 = r565803 ? r565809 : r565832;
        double r565834 = r565789 ? r565801 : r565833;
        return r565834;
}

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
Target5.5
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (* z 9.0) t) < -4.479651987078281e+284

    1. Initial program 53.7

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

    3. Applied div-sub53.7

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

    4. Simplified53.7

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

    5. Simplified7.9

      \[\leadsto \frac{\frac{x \cdot y}{2}}{a} - \color{blue}{\frac{9}{\frac{\frac{2 \cdot a}{t}}{z}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity7.9

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

    8. Applied *-un-lft-identity7.9

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

    9. Applied times-frac7.9

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

    10. Applied times-frac0.8

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

    11. Simplified0.8

      \[\leadsto \color{blue}{x} \cdot \frac{\frac{y}{2}}{a} - \frac{9}{\frac{\frac{2 \cdot a}{t}}{z}}\]

    if -4.479651987078281e+284 < (* (* z 9.0) t) < -8.458949463653347e-194 or 1.5365662849193131e-202 < (* (* z 9.0) t) < 1.5919249042077322e+162

    1. Initial program 3.6

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

    3. Applied sub-neg3.6

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

    4. Simplified3.6

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

    if -8.458949463653347e-194 < (* (* z 9.0) t) < 1.5365662849193131e-202

    1. Initial program 4.7

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

    2. Taylor expanded around 0 4.8

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

    3. Simplified6.6

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

    if 1.5919249042077322e+162 < (* (* z 9.0) t)

    1. Initial program 24.4

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

    3. Applied div-sub24.4

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

    4. Simplified24.4

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

    5. Simplified7.2

      \[\leadsto \frac{\frac{x \cdot y}{2}}{a} - \color{blue}{\frac{9}{\frac{\frac{2 \cdot a}{t}}{z}}}\]
    6. Using strategy rm

    7. Applied div-inv7.3

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

    8. Applied add-sqr-sqrt7.3

      \[\leadsto \frac{\frac{x \cdot y}{2}}{a} - \frac{\color{blue}{\sqrt{9} \cdot \sqrt{9}}}{\frac{2 \cdot a}{t} \cdot \frac{1}{z}}\]

    9. Applied times-frac7.6

      \[\leadsto \frac{\frac{x \cdot y}{2}}{a} - \color{blue}{\frac{\sqrt{9}}{\frac{2 \cdot a}{t}} \cdot \frac{\sqrt{9}}{\frac{1}{z}}}\]

    10. Simplified7.5

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

    11. Simplified7.4

      \[\leadsto \frac{\frac{x \cdot y}{2}}{a} - \left(\frac{\frac{\sqrt{9}}{2}}{a} \cdot t\right) \cdot \color{blue}{\left(\sqrt{9} \cdot z\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -4.479651987078281100945877712393720755092 \cdot 10^{284}:\\ \;\;\;\;\frac{\frac{y}{2}}{a} \cdot x - \frac{9}{\frac{\frac{a \cdot 2}{t}}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -8.458949463653347121961008011969069086603 \cdot 10^{-194}:\\ \;\;\;\;\frac{x \cdot y + \left(t \cdot z\right) \cdot \left(-9\right)}{a \cdot 2}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.536566284919313133153449110235493411241 \cdot 10^{-202}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{a}{y}} - \frac{4.5}{\frac{a}{t \cdot z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 1.591924904207732236217241191554461171523 \cdot 10^{162}:\\ \;\;\;\;\frac{x \cdot y + \left(t \cdot z\right) \cdot \left(-9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot y}{2}}{a} - \left(\frac{\frac{\sqrt{9}}{2}}{a} \cdot t\right) \cdot \left(z \cdot \sqrt{9}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :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 (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))