Average Error: 7.2 → 0.8
Time: 16.9s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9.0\right) \cdot t}{a \cdot 2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9.0\right) \cdot t = -\infty:\\ \;\;\;\;\left(\frac{y}{a} \cdot x\right) \cdot 0.5 - \frac{z}{a} \cdot \left(t \cdot 4.5\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9.0\right) \cdot t \le 1.553517178096751 \cdot 10^{+221}:\\ \;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} \cdot x\right) \cdot 0.5 - \frac{z}{a} \cdot \left(t \cdot 4.5\right)\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9.0\right) \cdot t}{a \cdot 2.0}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9.0\right) \cdot t = -\infty:\\
\;\;\;\;\left(\frac{y}{a} \cdot x\right) \cdot 0.5 - \frac{z}{a} \cdot \left(t \cdot 4.5\right)\\

\mathbf{elif}\;x \cdot y - \left(z \cdot 9.0\right) \cdot t \le 1.553517178096751 \cdot 10^{+221}:\\
\;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r14207967 = x;
        double r14207968 = y;
        double r14207969 = r14207967 * r14207968;
        double r14207970 = z;
        double r14207971 = 9.0;
        double r14207972 = r14207970 * r14207971;
        double r14207973 = t;
        double r14207974 = r14207972 * r14207973;
        double r14207975 = r14207969 - r14207974;
        double r14207976 = a;
        double r14207977 = 2.0;
        double r14207978 = r14207976 * r14207977;
        double r14207979 = r14207975 / r14207978;
        return r14207979;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14207980 = x;
        double r14207981 = y;
        double r14207982 = r14207980 * r14207981;
        double r14207983 = z;
        double r14207984 = 9.0;
        double r14207985 = r14207983 * r14207984;
        double r14207986 = t;
        double r14207987 = r14207985 * r14207986;
        double r14207988 = r14207982 - r14207987;
        double r14207989 = -inf.0;
        bool r14207990 = r14207988 <= r14207989;
        double r14207991 = a;
        double r14207992 = r14207981 / r14207991;
        double r14207993 = r14207992 * r14207980;
        double r14207994 = 0.5;
        double r14207995 = r14207993 * r14207994;
        double r14207996 = r14207983 / r14207991;
        double r14207997 = 4.5;
        double r14207998 = r14207986 * r14207997;
        double r14207999 = r14207996 * r14207998;
        double r14208000 = r14207995 - r14207999;
        double r14208001 = 1.553517178096751e+221;
        bool r14208002 = r14207988 <= r14208001;
        double r14208003 = r14207982 / r14207991;
        double r14208004 = r14208003 * r14207994;
        double r14208005 = r14207983 * r14207986;
        double r14208006 = r14208005 * r14207997;
        double r14208007 = r14208006 / r14207991;
        double r14208008 = r14208004 - r14208007;
        double r14208009 = r14208002 ? r14208008 : r14208000;
        double r14208010 = r14207990 ? r14208000 : r14208009;
        return r14208010;
}

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

Original7.2
Target5.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9.0 \cdot t\right)}{a \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 1.553517178096751e+221 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 40.7

      \[\frac{x \cdot y - \left(z \cdot 9.0\right) \cdot t}{a \cdot 2.0}\]

    2. Taylor expanded around 0 40.3

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

    4. Applied *-un-lft-identity40.3

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

    5. Applied times-frac20.4

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

    6. Simplified20.4

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

    8. Applied *-un-lft-identity20.4

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

    9. Applied times-frac0.7

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

    10. Applied associate-*r*0.8

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

    11. Simplified0.8

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

    if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 1.553517178096751e+221

    1. Initial program 0.8

      \[\frac{x \cdot y - \left(z \cdot 9.0\right) \cdot t}{a \cdot 2.0}\]

    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

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9.0\right) \cdot t = -\infty:\\ \;\;\;\;\left(\frac{y}{a} \cdot x\right) \cdot 0.5 - \frac{z}{a} \cdot \left(t \cdot 4.5\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9.0\right) \cdot t \le 1.553517178096751 \cdot 10^{+221}:\\ \;\;\;\;\frac{x \cdot y}{a} \cdot 0.5 - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} \cdot x\right) \cdot 0.5 - \frac{z}{a} \cdot \left(t \cdot 4.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))