Average Error: 7.8 → 5.1
Time: 4.3s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.019204525666223946553497990031878832169 \cdot 10^{163} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.585567167112418833993175021914148886831 \cdot 10^{159}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{2} \cdot \frac{z - \left(\left(-z\right) + z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.019204525666223946553497990031878832169 \cdot 10^{163} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.585567167112418833993175021914148886831 \cdot 10^{159}\right):\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{2} \cdot \frac{z - \left(\left(-z\right) + z\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r851755 = x;
        double r851756 = y;
        double r851757 = r851755 * r851756;
        double r851758 = z;
        double r851759 = 9.0;
        double r851760 = r851758 * r851759;
        double r851761 = t;
        double r851762 = r851760 * r851761;
        double r851763 = r851757 - r851762;
        double r851764 = a;
        double r851765 = 2.0;
        double r851766 = r851764 * r851765;
        double r851767 = r851763 / r851766;
        return r851767;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r851768 = x;
        double r851769 = y;
        double r851770 = r851768 * r851769;
        double r851771 = z;
        double r851772 = 9.0;
        double r851773 = r851771 * r851772;
        double r851774 = t;
        double r851775 = r851773 * r851774;
        double r851776 = r851770 - r851775;
        double r851777 = -4.019204525666224e+163;
        bool r851778 = r851776 <= r851777;
        double r851779 = 3.585567167112419e+159;
        bool r851780 = r851776 <= r851779;
        double r851781 = !r851780;
        bool r851782 = r851778 || r851781;
        double r851783 = a;
        double r851784 = 2.0;
        double r851785 = r851783 * r851784;
        double r851786 = r851770 / r851785;
        double r851787 = r851772 * r851774;
        double r851788 = r851787 / r851784;
        double r851789 = -r851771;
        double r851790 = r851789 + r851771;
        double r851791 = r851771 - r851790;
        double r851792 = r851791 / r851783;
        double r851793 = r851788 * r851792;
        double r851794 = r851786 - r851793;
        double r851795 = 1.0;
        double r851796 = r851785 / r851776;
        double r851797 = r851795 / r851796;
        double r851798 = r851782 ? r851794 : r851797;
        return r851798;
}

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
Target5.4
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 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -4.019204525666224e+163 or 3.585567167112419e+159 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 22.8

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

    3. Applied associate-*l*22.7

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

    5. Applied prod-diff22.7

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

    6. Simplified22.7

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

    7. Simplified22.7

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

    9. Applied associate-+l-22.7

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

    10. Applied div-sub22.7

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

    11. Simplified13.5

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

    if -4.019204525666224e+163 < (- (* x y) (* (* z 9.0) t)) < 3.585567167112419e+159

    1. Initial program 1.0

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

    3. Applied clear-num1.3

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.019204525666223946553497990031878832169 \cdot 10^{163} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.585567167112418833993175021914148886831 \cdot 10^{159}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{2} \cdot \frac{z - \left(\left(-z\right) + z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))