Average Error: 8.0 → 1.7
Time: 4.6s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.81880234558190669 \cdot 10^{134} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.95624520253703639 \cdot 10^{220}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]
\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 -5.81880234558190669 \cdot 10^{134} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.95624520253703639 \cdot 10^{220}\right):\\
\;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r684227 = x;
        double r684228 = y;
        double r684229 = r684227 * r684228;
        double r684230 = z;
        double r684231 = 9.0;
        double r684232 = r684230 * r684231;
        double r684233 = t;
        double r684234 = r684232 * r684233;
        double r684235 = r684229 - r684234;
        double r684236 = a;
        double r684237 = 2.0;
        double r684238 = r684236 * r684237;
        double r684239 = r684235 / r684238;
        return r684239;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r684240 = x;
        double r684241 = y;
        double r684242 = r684240 * r684241;
        double r684243 = z;
        double r684244 = 9.0;
        double r684245 = r684243 * r684244;
        double r684246 = t;
        double r684247 = r684245 * r684246;
        double r684248 = r684242 - r684247;
        double r684249 = -5.818802345581907e+134;
        bool r684250 = r684248 <= r684249;
        double r684251 = 1.9562452025370364e+220;
        bool r684252 = r684248 <= r684251;
        double r684253 = !r684252;
        bool r684254 = r684250 || r684253;
        double r684255 = 0.5;
        double r684256 = r684240 * r684255;
        double r684257 = a;
        double r684258 = r684241 / r684257;
        double r684259 = r684256 * r684258;
        double r684260 = 4.5;
        double r684261 = r684246 * r684260;
        double r684262 = r684243 / r684257;
        double r684263 = r684261 * r684262;
        double r684264 = r684259 - r684263;
        double r684265 = 1.0;
        double r684266 = r684265 / r684257;
        double r684267 = 2.0;
        double r684268 = r684248 / r684267;
        double r684269 = r684266 * r684268;
        double r684270 = r684254 ? r684264 : r684269;
        return r684270;
}

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
Target5.6
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -5.818802345581907e+134 or 1.9562452025370364e+220 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 24.6

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

    2. Taylor expanded around 0 24.2

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

    4. Applied *-un-lft-identity24.2

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

    5. Applied times-frac14.1

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

    6. Applied associate-*r*14.2

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

    7. Simplified14.2

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

    9. Applied *-un-lft-identity14.2

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

    10. Applied times-frac2.9

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

    11. Applied associate-*r*2.8

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

    12. Simplified2.8

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

    if -5.818802345581907e+134 < (- (* x y) (* (* z 9.0) t)) < 1.9562452025370364e+220

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]

    4. Applied times-frac1.2

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.81880234558190669 \cdot 10^{134} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.95624520253703639 \cdot 10^{220}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))