Average Error: 8.0 → 1.1
Time: 5.4s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} = -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \le 3.7074446346728667 \cdot 10^{287}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} = -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \le 3.7074446346728667 \cdot 10^{287}\right):\\
\;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r836358 = x;
        double r836359 = y;
        double r836360 = r836358 * r836359;
        double r836361 = z;
        double r836362 = 9.0;
        double r836363 = r836361 * r836362;
        double r836364 = t;
        double r836365 = r836363 * r836364;
        double r836366 = r836360 - r836365;
        double r836367 = a;
        double r836368 = 2.0;
        double r836369 = r836367 * r836368;
        double r836370 = r836366 / r836369;
        return r836370;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r836371 = x;
        double r836372 = y;
        double r836373 = r836371 * r836372;
        double r836374 = z;
        double r836375 = 9.0;
        double r836376 = r836374 * r836375;
        double r836377 = t;
        double r836378 = r836376 * r836377;
        double r836379 = r836373 - r836378;
        double r836380 = a;
        double r836381 = 2.0;
        double r836382 = r836380 * r836381;
        double r836383 = r836379 / r836382;
        double r836384 = -inf.0;
        bool r836385 = r836383 <= r836384;
        double r836386 = 3.707444634672867e+287;
        bool r836387 = r836383 <= r836386;
        double r836388 = !r836387;
        bool r836389 = r836385 || r836388;
        double r836390 = 0.5;
        double r836391 = cbrt(r836380);
        double r836392 = r836391 * r836391;
        double r836393 = r836371 / r836392;
        double r836394 = r836372 / r836391;
        double r836395 = r836393 * r836394;
        double r836396 = r836390 * r836395;
        double r836397 = 4.5;
        double r836398 = r836374 / r836380;
        double r836399 = r836377 * r836398;
        double r836400 = r836397 * r836399;
        double r836401 = r836396 - r836400;
        double r836402 = r836373 / r836380;
        double r836403 = r836390 * r836402;
        double r836404 = r836377 * r836374;
        double r836405 = r836397 * r836404;
        double r836406 = 1.0;
        double r836407 = r836406 / r836380;
        double r836408 = r836405 * r836407;
        double r836409 = r836403 - r836408;
        double r836410 = r836389 ? r836401 : r836409;
        return r836410;
}

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

Original8.0
Target5.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)) < -inf.0 or 3.707444634672867e+287 < (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))

    1. Initial program 57.8

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

    2. Taylor expanded around 0 57.2

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

    4. Applied add-cube-cbrt57.3

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

    5. Applied times-frac31.7

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

    7. Applied *-un-lft-identity31.7

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

    8. Applied times-frac2.9

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

    9. Simplified2.9

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

    if -inf.0 < (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)) < 3.707444634672867e+287

    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 div-inv0.8

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

    5. Applied associate-*r*0.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} = -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \le 3.7074446346728667 \cdot 10^{287}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(4.5 \cdot \left(t \cdot z\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))