Average Error: 7.4 → 5.1
Time: 23.0s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -9.660763965508467762144727294322758643538 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot 4.5\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;x \cdot y \le -5.957190730987083552549836165375073036366 \cdot 10^{-220}:\\ \;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \sqrt{4.5} \cdot \left(\sqrt{4.5} \cdot \frac{t \cdot z}{a}\right)\\ \mathbf{elif}\;x \cdot y \le 1.75883717865655323524967542350152746508 \cdot 10^{-268}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \le 2.508463496121614942709578561102143380877 \cdot 10^{198}:\\ \;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \sqrt{4.5} \cdot \left(\sqrt{4.5} \cdot \frac{t \cdot z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -9.660763965508467762144727294322758643538 \cdot 10^{-31}:\\
\;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot 4.5\right) \cdot \frac{z}{\sqrt[3]{a}}\\

\mathbf{elif}\;x \cdot y \le -5.957190730987083552549836165375073036366 \cdot 10^{-220}:\\
\;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \sqrt{4.5} \cdot \left(\sqrt{4.5} \cdot \frac{t \cdot z}{a}\right)\\

\mathbf{elif}\;x \cdot y \le 1.75883717865655323524967542350152746508 \cdot 10^{-268}:\\
\;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\

\mathbf{elif}\;x \cdot y \le 2.508463496121614942709578561102143380877 \cdot 10^{198}:\\
\;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \sqrt{4.5} \cdot \left(\sqrt{4.5} \cdot \frac{t \cdot z}{a}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r31534726 = x;
        double r31534727 = y;
        double r31534728 = r31534726 * r31534727;
        double r31534729 = z;
        double r31534730 = 9.0;
        double r31534731 = r31534729 * r31534730;
        double r31534732 = t;
        double r31534733 = r31534731 * r31534732;
        double r31534734 = r31534728 - r31534733;
        double r31534735 = a;
        double r31534736 = 2.0;
        double r31534737 = r31534735 * r31534736;
        double r31534738 = r31534734 / r31534737;
        return r31534738;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r31534739 = x;
        double r31534740 = y;
        double r31534741 = r31534739 * r31534740;
        double r31534742 = -9.660763965508468e-31;
        bool r31534743 = r31534741 <= r31534742;
        double r31534744 = a;
        double r31534745 = r31534744 / r31534740;
        double r31534746 = r31534739 / r31534745;
        double r31534747 = 0.5;
        double r31534748 = r31534746 * r31534747;
        double r31534749 = t;
        double r31534750 = cbrt(r31534744);
        double r31534751 = r31534750 * r31534750;
        double r31534752 = r31534749 / r31534751;
        double r31534753 = 4.5;
        double r31534754 = r31534752 * r31534753;
        double r31534755 = z;
        double r31534756 = r31534755 / r31534750;
        double r31534757 = r31534754 * r31534756;
        double r31534758 = r31534748 - r31534757;
        double r31534759 = -5.9571907309870836e-220;
        bool r31534760 = r31534741 <= r31534759;
        double r31534761 = r31534741 / r31534744;
        double r31534762 = r31534761 * r31534747;
        double r31534763 = sqrt(r31534753);
        double r31534764 = r31534749 * r31534755;
        double r31534765 = r31534764 / r31534744;
        double r31534766 = r31534763 * r31534765;
        double r31534767 = r31534763 * r31534766;
        double r31534768 = r31534762 - r31534767;
        double r31534769 = 1.7588371786565532e-268;
        bool r31534770 = r31534741 <= r31534769;
        double r31534771 = r31534753 * r31534749;
        double r31534772 = r31534755 / r31534744;
        double r31534773 = r31534771 * r31534772;
        double r31534774 = r31534748 - r31534773;
        double r31534775 = 2.508463496121615e+198;
        bool r31534776 = r31534741 <= r31534775;
        double r31534777 = r31534776 ? r31534768 : r31534774;
        double r31534778 = r31534770 ? r31534774 : r31534777;
        double r31534779 = r31534760 ? r31534768 : r31534778;
        double r31534780 = r31534743 ? r31534758 : r31534779;
        return r31534780;
}

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

Original7.4
Target5.4
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -9.660763965508468e-31

    1. Initial program 11.3

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

    2. Taylor expanded around 0 11.2

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

    4. Applied associate-/l*10.1

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

    6. Applied add-cube-cbrt10.3

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    7. Applied times-frac8.0

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \color{blue}{\left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)}\]

    8. Applied associate-*r*8.0

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}}\]

    if -9.660763965508468e-31 < (* x y) < -5.9571907309870836e-220 or 1.7588371786565532e-268 < (* x y) < 2.508463496121615e+198

    1. Initial program 3.5

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

    2. Taylor expanded around 0 3.5

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

    4. Applied add-sqr-sqrt3.8

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

    5. Applied associate-*l*3.8

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

    if -5.9571907309870836e-220 < (* x y) < 1.7588371786565532e-268 or 2.508463496121615e+198 < (* x y)

    1. Initial program 10.7

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

    2. Taylor expanded around 0 10.7

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

    4. Applied associate-/l*4.9

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

    6. Applied *-un-lft-identity4.9

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

    7. Applied times-frac5.0

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

    8. Applied associate-*r*5.0

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

    9. Simplified5.0

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -9.660763965508467762144727294322758643538 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot 4.5\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;x \cdot y \le -5.957190730987083552549836165375073036366 \cdot 10^{-220}:\\ \;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \sqrt{4.5} \cdot \left(\sqrt{4.5} \cdot \frac{t \cdot z}{a}\right)\\ \mathbf{elif}\;x \cdot y \le 1.75883717865655323524967542350152746508 \cdot 10^{-268}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \le 2.508463496121614942709578561102143380877 \cdot 10^{198}:\\ \;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \sqrt{4.5} \cdot \left(\sqrt{4.5} \cdot \frac{t \cdot z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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)))