Average Error: 7.7 → 4.2
Time: 4.3s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le -3.21071016327233009 \cdot 10^{-73}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \le 1.37889474371042372 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{1}{\frac{a}{t \cdot z}}\\ \mathbf{elif}\;x \cdot y \le 1.93687473196227216 \cdot 10^{296}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y = -\infty:\\
\;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\

\mathbf{elif}\;x \cdot y \le -3.21071016327233009 \cdot 10^{-73}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\

\mathbf{elif}\;x \cdot y \le 1.37889474371042372 \cdot 10^{-21}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{1}{\frac{a}{t \cdot z}}\\

\mathbf{elif}\;x \cdot y \le 1.93687473196227216 \cdot 10^{296}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r856313 = x;
        double r856314 = y;
        double r856315 = r856313 * r856314;
        double r856316 = z;
        double r856317 = 9.0;
        double r856318 = r856316 * r856317;
        double r856319 = t;
        double r856320 = r856318 * r856319;
        double r856321 = r856315 - r856320;
        double r856322 = a;
        double r856323 = 2.0;
        double r856324 = r856322 * r856323;
        double r856325 = r856321 / r856324;
        return r856325;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r856326 = x;
        double r856327 = y;
        double r856328 = r856326 * r856327;
        double r856329 = -inf.0;
        bool r856330 = r856328 <= r856329;
        double r856331 = 0.5;
        double r856332 = r856326 * r856331;
        double r856333 = a;
        double r856334 = r856327 / r856333;
        double r856335 = r856332 * r856334;
        double r856336 = 4.5;
        double r856337 = t;
        double r856338 = z;
        double r856339 = r856337 * r856338;
        double r856340 = r856339 / r856333;
        double r856341 = r856336 * r856340;
        double r856342 = r856335 - r856341;
        double r856343 = -3.21071016327233e-73;
        bool r856344 = r856328 <= r856343;
        double r856345 = r856328 / r856333;
        double r856346 = r856331 * r856345;
        double r856347 = r856337 * r856336;
        double r856348 = r856338 / r856333;
        double r856349 = r856347 * r856348;
        double r856350 = r856346 - r856349;
        double r856351 = 1.3788947437104237e-21;
        bool r856352 = r856328 <= r856351;
        double r856353 = 1.0;
        double r856354 = r856333 / r856339;
        double r856355 = r856353 / r856354;
        double r856356 = r856336 * r856355;
        double r856357 = r856346 - r856356;
        double r856358 = 1.936874731962272e+296;
        bool r856359 = r856328 <= r856358;
        double r856360 = r856333 / r856338;
        double r856361 = r856337 / r856360;
        double r856362 = r856336 * r856361;
        double r856363 = r856346 - r856362;
        double r856364 = r856359 ? r856363 : r856342;
        double r856365 = r856352 ? r856357 : r856364;
        double r856366 = r856344 ? r856350 : r856365;
        double r856367 = r856330 ? r856342 : r856366;
        return r856367;
}

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

Original7.7
Target5.5
Herbie4.2
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* x y) < -inf.0 or 1.936874731962272e+296 < (* x y)

    1. Initial program 61.4

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

    2. Taylor expanded around 0 61.4

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

    4. Applied *-un-lft-identity61.4

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

    5. Applied times-frac6.0

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

    6. Applied associate-*r*6.0

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

    7. Simplified6.0

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

    if -inf.0 < (* x y) < -3.21071016327233e-73

    1. Initial program 3.7

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

    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 *-un-lft-identity3.8

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

    5. Applied times-frac3.2

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

    6. Applied associate-*r*3.2

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

    7. Simplified3.2

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

    if -3.21071016327233e-73 < (* x y) < 1.3788947437104237e-21

    1. Initial program 4.8

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

    2. Taylor expanded around 0 4.8

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

    4. Applied clear-num5.1

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

    if 1.3788947437104237e-21 < (* x y) < 1.936874731962272e+296

    1. Initial program 3.6

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

    2. Taylor expanded around 0 3.4

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

    4. Applied associate-/l*2.6

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le -3.21071016327233009 \cdot 10^{-73}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \le 1.37889474371042372 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{1}{\frac{a}{t \cdot z}}\\ \mathbf{elif}\;x \cdot y \le 1.93687473196227216 \cdot 10^{296}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))