Average Error: 7.9 → 3.9
Time: 23.5s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -6.997492932900559610807387797805298705116 \cdot 10^{193}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{elif}\;x \cdot y \le 1.144427839264607317163410286721664133085 \cdot 10^{212}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5 - \left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -6.997492932900559610807387797805298705116 \cdot 10^{193}:\\
\;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\

\mathbf{elif}\;x \cdot y \le 1.144427839264607317163410286721664133085 \cdot 10^{212}:\\
\;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5 - \left(z \cdot t\right) \cdot 4.5}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r32880501 = x;
        double r32880502 = y;
        double r32880503 = r32880501 * r32880502;
        double r32880504 = z;
        double r32880505 = 9.0;
        double r32880506 = r32880504 * r32880505;
        double r32880507 = t;
        double r32880508 = r32880506 * r32880507;
        double r32880509 = r32880503 - r32880508;
        double r32880510 = a;
        double r32880511 = 2.0;
        double r32880512 = r32880510 * r32880511;
        double r32880513 = r32880509 / r32880512;
        return r32880513;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r32880514 = x;
        double r32880515 = y;
        double r32880516 = r32880514 * r32880515;
        double r32880517 = -6.99749293290056e+193;
        bool r32880518 = r32880516 <= r32880517;
        double r32880519 = 0.5;
        double r32880520 = a;
        double r32880521 = cbrt(r32880520);
        double r32880522 = r32880521 * r32880521;
        double r32880523 = r32880514 / r32880522;
        double r32880524 = r32880515 / r32880521;
        double r32880525 = r32880523 * r32880524;
        double r32880526 = r32880519 * r32880525;
        double r32880527 = t;
        double r32880528 = z;
        double r32880529 = r32880528 / r32880520;
        double r32880530 = r32880527 * r32880529;
        double r32880531 = 4.5;
        double r32880532 = r32880530 * r32880531;
        double r32880533 = r32880526 - r32880532;
        double r32880534 = 1.1444278392646073e+212;
        bool r32880535 = r32880516 <= r32880534;
        double r32880536 = r32880516 * r32880519;
        double r32880537 = r32880528 * r32880527;
        double r32880538 = r32880537 * r32880531;
        double r32880539 = r32880536 - r32880538;
        double r32880540 = r32880539 / r32880520;
        double r32880541 = r32880535 ? r32880540 : r32880533;
        double r32880542 = r32880518 ? r32880533 : r32880541;
        return r32880542;
}

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

Original7.9
Target5.7
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x y) < -6.99749293290056e+193 or 1.1444278392646073e+212 < (* x y)

    1. Initial program 31.1

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

    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 add-cube-cbrt31.7

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

    5. Applied times-frac7.5

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

    7. Applied *-un-lft-identity7.5

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

    8. Applied times-frac2.0

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

    9. Simplified2.0

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

    if -6.99749293290056e+193 < (* x y) < 1.1444278392646073e+212

    1. Initial program 4.2

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

    2. Taylor expanded around 0 4.1

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

    4. Applied associate-*r/4.1

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

    5. Applied associate-*r/4.1

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

    6. Applied sub-div4.1

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -6.997492932900559610807387797805298705116 \cdot 10^{193}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{elif}\;x \cdot y \le 1.144427839264607317163410286721664133085 \cdot 10^{212}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5 - \left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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)))