Average Error: 7.3 → 6.2
Time: 21.4s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9.0\right) \cdot t}{a \cdot 2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot 2.0 \le -8.452560769864224 \cdot 10^{+197}:\\ \;\;\;\;\left(x \cdot \frac{y}{a}\right) \cdot 0.5 - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2.0 \le 1.9902573105400113 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{elif}\;a \cdot 2.0 \le 3.2154246053472375 \cdot 10^{+277}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - \frac{t}{\frac{a}{z}} \cdot 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{y}{a}\right) \cdot 0.5 - 4.5 \cdot \frac{z \cdot t}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9.0\right) \cdot t}{a \cdot 2.0}
↓
\begin{array}{l}
\mathbf{if}\;a \cdot 2.0 \le -8.452560769864224 \cdot 10^{+197}:\\
\;\;\;\;\left(x \cdot \frac{y}{a}\right) \cdot 0.5 - 4.5 \cdot \frac{z \cdot t}{a}\\

\mathbf{elif}\;a \cdot 2.0 \le 1.9902573105400113 \cdot 10^{+77}:\\
\;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\

\mathbf{elif}\;a \cdot 2.0 \le 3.2154246053472375 \cdot 10^{+277}:\\
\;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - \frac{t}{\frac{a}{z}} \cdot 4.5\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r32379490 = x;
        double r32379491 = y;
        double r32379492 = r32379490 * r32379491;
        double r32379493 = z;
        double r32379494 = 9.0;
        double r32379495 = r32379493 * r32379494;
        double r32379496 = t;
        double r32379497 = r32379495 * r32379496;
        double r32379498 = r32379492 - r32379497;
        double r32379499 = a;
        double r32379500 = 2.0;
        double r32379501 = r32379499 * r32379500;
        double r32379502 = r32379498 / r32379501;
        return r32379502;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r32379503 = a;
        double r32379504 = 2.0;
        double r32379505 = r32379503 * r32379504;
        double r32379506 = -8.452560769864224e+197;
        bool r32379507 = r32379505 <= r32379506;
        double r32379508 = x;
        double r32379509 = y;
        double r32379510 = r32379509 / r32379503;
        double r32379511 = r32379508 * r32379510;
        double r32379512 = 0.5;
        double r32379513 = r32379511 * r32379512;
        double r32379514 = 4.5;
        double r32379515 = z;
        double r32379516 = t;
        double r32379517 = r32379515 * r32379516;
        double r32379518 = r32379517 / r32379503;
        double r32379519 = r32379514 * r32379518;
        double r32379520 = r32379513 - r32379519;
        double r32379521 = 1.9902573105400113e+77;
        bool r32379522 = r32379505 <= r32379521;
        double r32379523 = r32379509 * r32379508;
        double r32379524 = r32379523 / r32379503;
        double r32379525 = r32379512 * r32379524;
        double r32379526 = r32379517 * r32379514;
        double r32379527 = r32379526 / r32379503;
        double r32379528 = r32379525 - r32379527;
        double r32379529 = 3.2154246053472375e+277;
        bool r32379530 = r32379505 <= r32379529;
        double r32379531 = r32379503 / r32379515;
        double r32379532 = r32379516 / r32379531;
        double r32379533 = r32379532 * r32379514;
        double r32379534 = r32379525 - r32379533;
        double r32379535 = r32379530 ? r32379534 : r32379520;
        double r32379536 = r32379522 ? r32379528 : r32379535;
        double r32379537 = r32379507 ? r32379520 : r32379536;
        return r32379537;
}

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

Original7.3
Target5.5
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9.0 \cdot t\right)}{a \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a 2.0) < -8.452560769864224e+197 or 3.2154246053472375e+277 < (* a 2.0)

    1. Initial program 13.7

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

    2. Taylor expanded around 0 13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied times-frac10.9

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

    6. Simplified10.9

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

    if -8.452560769864224e+197 < (* a 2.0) < 1.9902573105400113e+77

    1. Initial program 3.8

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

    2. Taylor expanded around 0 3.8

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

    4. Applied associate-*r/3.8

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

    if 1.9902573105400113e+77 < (* a 2.0) < 3.2154246053472375e+277

    1. Initial program 12.7

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

    2. Taylor expanded around 0 12.7

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

    4. Applied associate-/l*9.8

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2.0 \le -8.452560769864224 \cdot 10^{+197}:\\ \;\;\;\;\left(x \cdot \frac{y}{a}\right) \cdot 0.5 - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2.0 \le 1.9902573105400113 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{elif}\;a \cdot 2.0 \le 3.2154246053472375 \cdot 10^{+277}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - \frac{t}{\frac{a}{z}} \cdot 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{y}{a}\right) \cdot 0.5 - 4.5 \cdot \frac{z \cdot t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))