Average Error: 7.9 → 0.5
Time: 4.3s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.570465535572884322699978790782884605691 \cdot 10^{-101} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.47440313991068531415406912842436761104 \cdot 10^{-267} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.212678920874577831195231744431104229608 \cdot 10^{275}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.570465535572884322699978790782884605691 \cdot 10^{-101} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.47440313991068531415406912842436761104 \cdot 10^{-267} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.212678920874577831195231744431104229608 \cdot 10^{275}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot 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 r701212 = x;
        double r701213 = y;
        double r701214 = r701212 * r701213;
        double r701215 = z;
        double r701216 = 9.0;
        double r701217 = r701215 * r701216;
        double r701218 = t;
        double r701219 = r701217 * r701218;
        double r701220 = r701214 - r701219;
        double r701221 = a;
        double r701222 = 2.0;
        double r701223 = r701221 * r701222;
        double r701224 = r701220 / r701223;
        return r701224;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r701225 = x;
        double r701226 = y;
        double r701227 = r701225 * r701226;
        double r701228 = z;
        double r701229 = 9.0;
        double r701230 = r701228 * r701229;
        double r701231 = t;
        double r701232 = r701230 * r701231;
        double r701233 = r701227 - r701232;
        double r701234 = -inf.0;
        bool r701235 = r701233 <= r701234;
        double r701236 = -1.5704655355728843e-101;
        bool r701237 = r701233 <= r701236;
        double r701238 = 1.4744031399106853e-267;
        bool r701239 = r701233 <= r701238;
        double r701240 = 6.212678920874578e+275;
        bool r701241 = r701233 <= r701240;
        double r701242 = !r701241;
        bool r701243 = r701239 || r701242;
        double r701244 = !r701243;
        bool r701245 = r701237 || r701244;
        double r701246 = !r701245;
        bool r701247 = r701235 || r701246;
        double r701248 = 0.5;
        double r701249 = a;
        double r701250 = r701249 / r701226;
        double r701251 = r701225 / r701250;
        double r701252 = r701248 * r701251;
        double r701253 = 4.5;
        double r701254 = r701253 * r701231;
        double r701255 = r701228 / r701249;
        double r701256 = r701254 * r701255;
        double r701257 = r701252 - r701256;
        double r701258 = r701227 / r701249;
        double r701259 = r701248 * r701258;
        double r701260 = r701231 * r701228;
        double r701261 = r701260 / r701249;
        double r701262 = r701253 * r701261;
        double r701263 = r701259 - r701262;
        double r701264 = r701247 ? r701257 : r701263;
        return r701264;
}

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

Original7.9
Target5.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or -1.5704655355728843e-101 < (- (* x y) (* (* z 9.0) t)) < 1.4744031399106853e-267 or 6.212678920874578e+275 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 33.8

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

    2. Taylor expanded around 0 33.5

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

    4. Applied *-un-lft-identity33.5

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

    5. Applied times-frac19.0

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

    6. Simplified19.0

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

    8. Applied associate-*r*19.1

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

    10. Applied associate-/l*1.3

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

    if -inf.0 < (- (* x y) (* (* z 9.0) t)) < -1.5704655355728843e-101 or 1.4744031399106853e-267 < (- (* x y) (* (* z 9.0) t)) < 6.212678920874578e+275

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.3

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

    4. Applied *-un-lft-identity0.3

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

    5. Applied times-frac5.4

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

    6. Simplified5.4

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

    8. Applied associate-*r*5.4

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

    9. Taylor expanded around 0 0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.570465535572884322699978790782884605691 \cdot 10^{-101} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.47440313991068531415406912842436761104 \cdot 10^{-267} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.212678920874577831195231744431104229608 \cdot 10^{275}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))