Average Error: 7.6 → 4.1
Time: 7.0s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.21834298495767539 \cdot 10^{236}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le -1.3603949856993749 \cdot 10^{34}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;x \cdot y \le 2.9090994917840058 \cdot 10^{187}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{1}{\frac{a}{t \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -2.21834298495767539 \cdot 10^{236}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\

\mathbf{elif}\;x \cdot y \le -1.3603949856993749 \cdot 10^{34}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\

\mathbf{elif}\;x \cdot y \le 2.9090994917840058 \cdot 10^{187}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{1}{\frac{a}{t \cdot 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 r797908 = x;
        double r797909 = y;
        double r797910 = r797908 * r797909;
        double r797911 = z;
        double r797912 = 9.0;
        double r797913 = r797911 * r797912;
        double r797914 = t;
        double r797915 = r797913 * r797914;
        double r797916 = r797910 - r797915;
        double r797917 = a;
        double r797918 = 2.0;
        double r797919 = r797917 * r797918;
        double r797920 = r797916 / r797919;
        return r797920;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r797921 = x;
        double r797922 = y;
        double r797923 = r797921 * r797922;
        double r797924 = -2.2183429849576754e+236;
        bool r797925 = r797923 <= r797924;
        double r797926 = 0.5;
        double r797927 = a;
        double r797928 = r797927 / r797922;
        double r797929 = r797921 / r797928;
        double r797930 = r797926 * r797929;
        double r797931 = 4.5;
        double r797932 = t;
        double r797933 = z;
        double r797934 = r797932 * r797933;
        double r797935 = r797934 / r797927;
        double r797936 = r797931 * r797935;
        double r797937 = r797930 - r797936;
        double r797938 = -1.360394985699375e+34;
        bool r797939 = r797923 <= r797938;
        double r797940 = r797923 / r797927;
        double r797941 = r797926 * r797940;
        double r797942 = cbrt(r797927);
        double r797943 = r797942 * r797942;
        double r797944 = r797932 / r797943;
        double r797945 = r797931 * r797944;
        double r797946 = r797933 / r797942;
        double r797947 = r797945 * r797946;
        double r797948 = r797941 - r797947;
        double r797949 = 2.9090994917840058e+187;
        bool r797950 = r797923 <= r797949;
        double r797951 = 1.0;
        double r797952 = r797927 / r797934;
        double r797953 = r797951 / r797952;
        double r797954 = r797931 * r797953;
        double r797955 = r797941 - r797954;
        double r797956 = r797950 ? r797955 : r797937;
        double r797957 = r797939 ? r797948 : r797956;
        double r797958 = r797925 ? r797937 : r797957;
        return r797958;
}

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

Original7.6
Target5.6
Herbie4.1
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -2.2183429849576754e+236 or 2.9090994917840058e+187 < (* x y)

    1. Initial program 31.3

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

    2. Taylor expanded around 0 31.1

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

    4. Applied associate-/l*6.4

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

    if -2.2183429849576754e+236 < (* x y) < -1.360394985699375e+34

    1. Initial program 5.1

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

    2. Taylor expanded around 0 4.9

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

    4. Applied add-cube-cbrt5.1

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

    5. Applied times-frac2.0

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

    6. Applied associate-*r*2.1

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

    if -1.360394985699375e+34 < (* x y) < 2.9090994917840058e+187

    1. Initial program 3.9

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

    2. Taylor expanded around 0 3.8

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

    4. Applied clear-num4.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\frac{1}{\frac{a}{t \cdot z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.21834298495767539 \cdot 10^{236}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le -1.3603949856993749 \cdot 10^{34}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;x \cdot y \le 2.9090994917840058 \cdot 10^{187}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{1}{\frac{a}{t \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))