Average Error: 7.4 → 6.1
Time: 17.4s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.934648177654955394021045567009889574846 \cdot 10^{201}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \left(t \cdot \frac{z}{a}\right)\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.632362310242604630969163503387641631979 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \sqrt[3]{4.5} \cdot \left(\left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right) \cdot \sqrt[3]{4.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a} - \frac{\left(t \cdot z\right) \cdot 4.5}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.934648177654955394021045567009889574846 \cdot 10^{201}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \left(t \cdot \frac{z}{a}\right)\right)\\

\mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.632362310242604630969163503387641631979 \cdot 10^{-50}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \sqrt[3]{4.5} \cdot \left(\left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right) \cdot \sqrt[3]{4.5}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r441408 = x;
        double r441409 = y;
        double r441410 = r441408 * r441409;
        double r441411 = z;
        double r441412 = 9.0;
        double r441413 = r441411 * r441412;
        double r441414 = t;
        double r441415 = r441413 * r441414;
        double r441416 = r441410 - r441415;
        double r441417 = a;
        double r441418 = 2.0;
        double r441419 = r441417 * r441418;
        double r441420 = r441416 / r441419;
        return r441420;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r441421 = x;
        double r441422 = y;
        double r441423 = r441421 * r441422;
        double r441424 = z;
        double r441425 = 9.0;
        double r441426 = r441424 * r441425;
        double r441427 = t;
        double r441428 = r441426 * r441427;
        double r441429 = r441423 - r441428;
        double r441430 = -5.934648177654955e+201;
        bool r441431 = r441429 <= r441430;
        double r441432 = 0.5;
        double r441433 = a;
        double r441434 = r441423 / r441433;
        double r441435 = r441432 * r441434;
        double r441436 = 4.5;
        double r441437 = cbrt(r441436);
        double r441438 = r441437 * r441437;
        double r441439 = r441424 / r441433;
        double r441440 = r441427 * r441439;
        double r441441 = r441437 * r441440;
        double r441442 = r441438 * r441441;
        double r441443 = r441435 - r441442;
        double r441444 = 2.6323623102426046e-50;
        bool r441445 = r441429 <= r441444;
        double r441446 = r441427 * r441424;
        double r441447 = r441446 / r441433;
        double r441448 = r441437 * r441447;
        double r441449 = r441448 * r441437;
        double r441450 = r441437 * r441449;
        double r441451 = r441435 - r441450;
        double r441452 = r441432 * r441421;
        double r441453 = r441422 / r441433;
        double r441454 = r441452 * r441453;
        double r441455 = r441446 * r441436;
        double r441456 = r441455 / r441433;
        double r441457 = r441454 - r441456;
        double r441458 = r441445 ? r441451 : r441457;
        double r441459 = r441431 ? r441443 : r441458;
        return r441459;
}

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

Original7.4
Target5.6
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -5.934648177654955e+201

    1. Initial program 28.9

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

    2. Taylor expanded around 0 28.7

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

    4. Applied add-cube-cbrt28.7

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

    5. Applied associate-*l*28.8

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

    7. Applied *-un-lft-identity28.8

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

    8. Applied times-frac15.4

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

    9. Simplified15.4

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

    if -5.934648177654955e+201 < (- (* x y) (* (* z 9.0) t)) < 2.6323623102426046e-50

    1. Initial program 1.3

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cube-cbrt1.3

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

    5. Applied associate-*l*1.3

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

    7. Applied associate-*l*1.4

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

    8. Simplified1.4

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

    if 2.6323623102426046e-50 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 9.0

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

    2. Taylor expanded around 0 8.9

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

    4. Applied associate-*r/8.9

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

    5. Simplified8.9

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

    7. Applied *-un-lft-identity8.9

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

    8. Applied times-frac9.5

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

    9. Applied associate-*r*9.4

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

    10. Simplified9.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.934648177654955394021045567009889574846 \cdot 10^{201}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \left(t \cdot \frac{z}{a}\right)\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.632362310242604630969163503387641631979 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \sqrt[3]{4.5} \cdot \left(\left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right) \cdot \sqrt[3]{4.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a} - \frac{\left(t \cdot z\right) \cdot 4.5}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.14403070783397609e99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))