Average Error: 7.7 → 0.9
Time: 4.4s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.47596775768044329 \cdot 10^{214} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.44327490436144402 \cdot 10^{217}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.47596775768044329 \cdot 10^{214} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.44327490436144402 \cdot 10^{217}\right):\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r711876 = x;
        double r711877 = y;
        double r711878 = r711876 * r711877;
        double r711879 = z;
        double r711880 = 9.0;
        double r711881 = r711879 * r711880;
        double r711882 = t;
        double r711883 = r711881 * r711882;
        double r711884 = r711878 - r711883;
        double r711885 = a;
        double r711886 = 2.0;
        double r711887 = r711885 * r711886;
        double r711888 = r711884 / r711887;
        return r711888;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r711889 = x;
        double r711890 = y;
        double r711891 = r711889 * r711890;
        double r711892 = z;
        double r711893 = 9.0;
        double r711894 = r711892 * r711893;
        double r711895 = t;
        double r711896 = r711894 * r711895;
        double r711897 = r711891 - r711896;
        double r711898 = -1.4759677576804433e+214;
        bool r711899 = r711897 <= r711898;
        double r711900 = 1.443274904361444e+217;
        bool r711901 = r711897 <= r711900;
        double r711902 = !r711901;
        bool r711903 = r711899 || r711902;
        double r711904 = 0.5;
        double r711905 = a;
        double r711906 = r711890 / r711905;
        double r711907 = r711889 * r711906;
        double r711908 = r711904 * r711907;
        double r711909 = 4.5;
        double r711910 = r711905 / r711892;
        double r711911 = r711895 / r711910;
        double r711912 = r711909 * r711911;
        double r711913 = r711908 - r711912;
        double r711914 = r711904 * r711891;
        double r711915 = r711895 * r711892;
        double r711916 = r711909 * r711915;
        double r711917 = r711914 - r711916;
        double r711918 = r711917 / r711905;
        double r711919 = r711903 ? r711913 : r711918;
        return r711919;
}

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

Original7.7
Target5.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -1.4759677576804433e+214 or 1.443274904361444e+217 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 31.2

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

    2. Taylor expanded around 0 30.5

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

    4. Applied associate-/l*16.8

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

    6. Applied *-un-lft-identity16.8

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

    7. Applied times-frac1.0

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

    8. Simplified1.0

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

    if -1.4759677576804433e+214 < (- (* x y) (* (* z 9.0) t)) < 1.443274904361444e+217

    1. Initial program 0.8

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

    2. Taylor expanded around 0 0.8

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

    4. Applied associate-*r/0.8

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

    5. Applied associate-*r/0.8

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

    6. Applied sub-div0.8

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.47596775768044329 \cdot 10^{214} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.44327490436144402 \cdot 10^{217}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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)))