Average Error: 7.5 → 6.1
Time: 15.1s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot 2 \le -5.526614624971197016874091463897007724665 \cdot 10^{247}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \cdot 2 \le -2.351629294748413366835826341287596823597 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;a \cdot 2 \le 4.940347943794864332566063191313488771801 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\ \mathbf{elif}\;a \cdot 2 \le 6.077447378939868238265068590780103982245 \cdot 10^{221}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;a \cdot 2 \le -5.526614624971197016874091463897007724665 \cdot 10^{247}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{elif}\;a \cdot 2 \le -2.351629294748413366835826341287596823597 \cdot 10^{-14}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\

\mathbf{elif}\;a \cdot 2 \le 4.940347943794864332566063191313488771801 \cdot 10^{-102}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\

\mathbf{elif}\;a \cdot 2 \le 6.077447378939868238265068590780103982245 \cdot 10^{221}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r473659 = x;
        double r473660 = y;
        double r473661 = r473659 * r473660;
        double r473662 = z;
        double r473663 = 9.0;
        double r473664 = r473662 * r473663;
        double r473665 = t;
        double r473666 = r473664 * r473665;
        double r473667 = r473661 - r473666;
        double r473668 = a;
        double r473669 = 2.0;
        double r473670 = r473668 * r473669;
        double r473671 = r473667 / r473670;
        return r473671;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r473672 = a;
        double r473673 = 2.0;
        double r473674 = r473672 * r473673;
        double r473675 = -5.526614624971197e+247;
        bool r473676 = r473674 <= r473675;
        double r473677 = 0.5;
        double r473678 = x;
        double r473679 = y;
        double r473680 = r473678 * r473679;
        double r473681 = r473680 / r473672;
        double r473682 = r473677 * r473681;
        double r473683 = 4.5;
        double r473684 = t;
        double r473685 = z;
        double r473686 = r473672 / r473685;
        double r473687 = r473684 / r473686;
        double r473688 = r473683 * r473687;
        double r473689 = r473682 - r473688;
        double r473690 = -2.3516292947484134e-14;
        bool r473691 = r473674 <= r473690;
        double r473692 = r473679 / r473672;
        double r473693 = r473678 * r473692;
        double r473694 = r473677 * r473693;
        double r473695 = r473684 * r473685;
        double r473696 = r473695 / r473672;
        double r473697 = r473683 * r473696;
        double r473698 = r473694 - r473697;
        double r473699 = 4.940347943794864e-102;
        bool r473700 = r473674 <= r473699;
        double r473701 = 1.0;
        double r473702 = 9.0;
        double r473703 = r473685 * r473702;
        double r473704 = r473703 * r473684;
        double r473705 = r473680 - r473704;
        double r473706 = r473674 / r473705;
        double r473707 = r473701 / r473706;
        double r473708 = 6.077447378939868e+221;
        bool r473709 = r473674 <= r473708;
        double r473710 = r473672 / r473679;
        double r473711 = r473678 / r473710;
        double r473712 = r473677 * r473711;
        double r473713 = r473712 - r473697;
        double r473714 = r473709 ? r473689 : r473713;
        double r473715 = r473700 ? r473707 : r473714;
        double r473716 = r473691 ? r473698 : r473715;
        double r473717 = r473676 ? r473689 : r473716;
        return r473717;
}

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
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a 2.0) < -5.526614624971197e+247 or 4.940347943794864e-102 < (* a 2.0) < 6.077447378939868e+221

    1. Initial program 8.1

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

    2. Taylor expanded around 0 8.0

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

    4. Applied associate-/l*6.9

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

    if -5.526614624971197e+247 < (* a 2.0) < -2.3516292947484134e-14

    1. Initial program 9.6

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

    2. Taylor expanded around 0 9.5

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

    4. Applied *-un-lft-identity9.5

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

    5. Applied times-frac7.3

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

    6. Simplified7.3

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

    if -2.3516292947484134e-14 < (* a 2.0) < 4.940347943794864e-102

    1. Initial program 1.8

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

    3. Applied clear-num1.9

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

    if 6.077447378939868e+221 < (* a 2.0)

    1. Initial program 15.2

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

    2. Taylor expanded around 0 14.9

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

    4. Applied associate-/l*11.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \le -5.526614624971197016874091463897007724665 \cdot 10^{247}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \cdot 2 \le -2.351629294748413366835826341287596823597 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;a \cdot 2 \le 4.940347943794864332566063191313488771801 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\ \mathbf{elif}\;a \cdot 2 \le 6.077447378939868238265068590780103982245 \cdot 10^{221}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))