Average Error: 8.0 → 4.2
Time: 4.6s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -1.8850835762976409 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.4718778096067223 \cdot 10^{-110}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.2619286693938016 \cdot 10^{300}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]
\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} - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -1.8850835762976409 \cdot 10^{-58}:\\
\;\;\;\;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 2.4718778096067223 \cdot 10^{-110}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.2619286693938016 \cdot 10^{300}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 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 r857782 = x;
        double r857783 = y;
        double r857784 = r857782 * r857783;
        double r857785 = z;
        double r857786 = 9.0;
        double r857787 = r857785 * r857786;
        double r857788 = t;
        double r857789 = r857787 * r857788;
        double r857790 = r857784 - r857789;
        double r857791 = a;
        double r857792 = 2.0;
        double r857793 = r857791 * r857792;
        double r857794 = r857790 / r857793;
        return r857794;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r857795 = z;
        double r857796 = 9.0;
        double r857797 = r857795 * r857796;
        double r857798 = t;
        double r857799 = r857797 * r857798;
        double r857800 = -inf.0;
        bool r857801 = r857799 <= r857800;
        double r857802 = 0.5;
        double r857803 = x;
        double r857804 = y;
        double r857805 = r857803 * r857804;
        double r857806 = a;
        double r857807 = r857805 / r857806;
        double r857808 = r857802 * r857807;
        double r857809 = 4.5;
        double r857810 = r857795 / r857806;
        double r857811 = r857798 * r857810;
        double r857812 = r857809 * r857811;
        double r857813 = r857808 - r857812;
        double r857814 = -1.885083576297641e-58;
        bool r857815 = r857799 <= r857814;
        double r857816 = r857804 / r857806;
        double r857817 = r857803 * r857816;
        double r857818 = r857802 * r857817;
        double r857819 = r857798 * r857795;
        double r857820 = r857819 / r857806;
        double r857821 = r857809 * r857820;
        double r857822 = r857818 - r857821;
        double r857823 = 2.4718778096067223e-110;
        bool r857824 = r857799 <= r857823;
        double r857825 = cbrt(r857806);
        double r857826 = r857825 * r857825;
        double r857827 = r857798 / r857826;
        double r857828 = r857795 / r857825;
        double r857829 = r857827 * r857828;
        double r857830 = r857809 * r857829;
        double r857831 = r857808 - r857830;
        double r857832 = 2.2619286693938016e+300;
        bool r857833 = r857799 <= r857832;
        double r857834 = r857806 / r857804;
        double r857835 = r857803 / r857834;
        double r857836 = r857802 * r857835;
        double r857837 = r857836 - r857821;
        double r857838 = r857798 * r857809;
        double r857839 = r857838 * r857810;
        double r857840 = r857808 - r857839;
        double r857841 = r857833 ? r857837 : r857840;
        double r857842 = r857824 ? r857831 : r857841;
        double r857843 = r857815 ? r857822 : r857842;
        double r857844 = r857801 ? r857813 : r857843;
        return r857844;
}

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

Original8.0
Target5.8
Herbie4.2
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if (* (* z 9.0) t) < -inf.0

    1. Initial program 64.0

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

    2. Taylor expanded around 0 62.5

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

    4. Applied *-un-lft-identity62.5

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

    5. Applied times-frac5.5

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

    6. Simplified5.5

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

    if -inf.0 < (* (* z 9.0) t) < -1.885083576297641e-58

    1. Initial program 4.3

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

    2. Taylor expanded around 0 4.3

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

    4. Applied *-un-lft-identity4.3

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

    5. Applied times-frac3.1

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

    6. Simplified3.1

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

    if -1.885083576297641e-58 < (* (* z 9.0) t) < 2.4718778096067223e-110

    1. Initial program 4.4

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

    2. Taylor expanded around 0 4.4

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

    4. Applied add-cube-cbrt4.5

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

    5. Applied times-frac4.9

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

    if 2.4718778096067223e-110 < (* (* z 9.0) t) < 2.2619286693938016e+300

    1. Initial program 3.6

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

    2. Taylor expanded around 0 3.6

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

    4. Applied associate-/l*3.4

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

    if 2.2619286693938016e+300 < (* (* z 9.0) t)

    1. Initial program 60.1

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

    2. Taylor expanded around 0 58.7

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

    4. Applied *-un-lft-identity58.7

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

    5. Applied times-frac7.6

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

    6. Applied associate-*r*7.7

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

    7. Simplified7.7

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(t \cdot 4.5\right)} \cdot \frac{z}{a}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t = -\infty:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -1.8850835762976409 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.4718778096067223 \cdot 10^{-110}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.2619286693938016 \cdot 10^{300}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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)))