Average Error: 8.1 → 4.8
Time: 22.4s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{a}{y}} - \left(z \cdot \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right)\right) \cdot \left(\frac{\sqrt[3]{4.5}}{a} \cdot t\right)\\ \mathbf{elif}\;x \cdot y \le -4.132036820116678551692351105979607575537 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y}{2 \cdot a} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot x}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}} - \frac{4.5}{\frac{\frac{a}{z}}{t}}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y = -\infty:\\
\;\;\;\;\frac{0.5 \cdot x}{\frac{a}{y}} - \left(z \cdot \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right)\right) \cdot \left(\frac{\sqrt[3]{4.5}}{a} \cdot t\right)\\

\mathbf{elif}\;x \cdot y \le -4.132036820116678551692351105979607575537 \cdot 10^{-142}:\\
\;\;\;\;\frac{x \cdot y}{2 \cdot a} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{0.5 \cdot x}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}} - \frac{4.5}{\frac{\frac{a}{z}}{t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r609005 = x;
        double r609006 = y;
        double r609007 = r609005 * r609006;
        double r609008 = z;
        double r609009 = 9.0;
        double r609010 = r609008 * r609009;
        double r609011 = t;
        double r609012 = r609010 * r609011;
        double r609013 = r609007 - r609012;
        double r609014 = a;
        double r609015 = 2.0;
        double r609016 = r609014 * r609015;
        double r609017 = r609013 / r609016;
        return r609017;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r609018 = x;
        double r609019 = y;
        double r609020 = r609018 * r609019;
        double r609021 = -inf.0;
        bool r609022 = r609020 <= r609021;
        double r609023 = 0.5;
        double r609024 = r609023 * r609018;
        double r609025 = a;
        double r609026 = r609025 / r609019;
        double r609027 = r609024 / r609026;
        double r609028 = z;
        double r609029 = 4.5;
        double r609030 = cbrt(r609029);
        double r609031 = r609030 * r609030;
        double r609032 = r609028 * r609031;
        double r609033 = r609030 / r609025;
        double r609034 = t;
        double r609035 = r609033 * r609034;
        double r609036 = r609032 * r609035;
        double r609037 = r609027 - r609036;
        double r609038 = -4.1320368201166786e-142;
        bool r609039 = r609020 <= r609038;
        double r609040 = 2.0;
        double r609041 = r609040 * r609025;
        double r609042 = r609020 / r609041;
        double r609043 = 9.0;
        double r609044 = r609043 * r609034;
        double r609045 = r609044 / r609025;
        double r609046 = r609028 / r609040;
        double r609047 = r609045 * r609046;
        double r609048 = r609042 - r609047;
        double r609049 = cbrt(r609025);
        double r609050 = r609049 * r609049;
        double r609051 = cbrt(r609019);
        double r609052 = r609051 * r609051;
        double r609053 = r609050 / r609052;
        double r609054 = r609024 / r609053;
        double r609055 = r609049 / r609051;
        double r609056 = r609054 / r609055;
        double r609057 = r609025 / r609028;
        double r609058 = r609057 / r609034;
        double r609059 = r609029 / r609058;
        double r609060 = r609056 - r609059;
        double r609061 = r609039 ? r609048 : r609060;
        double r609062 = r609022 ? r609037 : r609061;
        return r609062;
}

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

Original8.1
Target5.5
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -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 64.0

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

    3. Simplified5.1

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

    5. Applied *-un-lft-identity5.1

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

    6. Applied times-frac0.3

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

    7. Applied add-cube-cbrt0.3

      \[\leadsto \frac{0.5 \cdot x}{\frac{a}{y}} - \frac{\color{blue}{\left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \sqrt[3]{4.5}}}{\frac{1}{z} \cdot \frac{a}{t}}\]

    8. Applied times-frac0.3

      \[\leadsto \frac{0.5 \cdot x}{\frac{a}{y}} - \color{blue}{\frac{\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}}{\frac{1}{z}} \cdot \frac{\sqrt[3]{4.5}}{\frac{a}{t}}}\]

    9. Simplified0.3

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

    10. Simplified0.3

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

    if -inf.0 < (* x y) < -4.1320368201166786e-142

    1. Initial program 4.2

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

    3. Applied div-sub4.2

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

    4. Simplified4.2

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

    5. Simplified4.2

      \[\leadsto \frac{x \cdot y}{2 \cdot a} - \color{blue}{\frac{z}{2} \cdot \frac{t \cdot 9}{a}}\]

    if -4.1320368201166786e-142 < (* x y)

    1. Initial program 7.4

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

    2. Taylor expanded around 0 7.3

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

    3. Simplified7.3

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

    5. Applied associate-/r*7.2

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

    7. Applied add-cube-cbrt7.5

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

    8. Applied add-cube-cbrt7.6

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

    9. Applied times-frac7.6

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

    10. Applied associate-/r*5.3

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{a}{y}} - \left(z \cdot \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right)\right) \cdot \left(\frac{\sqrt[3]{4.5}}{a} \cdot t\right)\\ \mathbf{elif}\;x \cdot y \le -4.132036820116678551692351105979607575537 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y}{2 \cdot a} - \frac{9 \cdot t}{a} \cdot \frac{z}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot x}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}} - \frac{4.5}{\frac{\frac{a}{z}}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :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 (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))