Average Error: 8.0 → 2.2
Time: 10.1s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -3.8035771558501627 \cdot 10^{137} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.046942143327783 \cdot 10^{126}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t} \cdot \sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}{\frac{a \cdot 2}{\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}}\\ \end{array}\]
\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 -3.8035771558501627 \cdot 10^{137} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.046942143327783 \cdot 10^{126}\right):\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t} \cdot \sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}{\frac{a \cdot 2}{\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}}\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) / ((double) (a * 2.0))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= -3.803577155850163e+137) || !(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 1.046942143327783e+126))) {
		VAR = ((double) (((double) (0.5 * ((double) (x * ((double) (y / a)))))) - ((double) (4.5 * ((double) (t / ((double) (a / z))))))));
	} else {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))))) * ((double) cbrt(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))))))) / ((double) (((double) (a * 2.0)) / ((double) cbrt(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t))))))))));
	}
	return VAR;
}

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
Target6.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -3.8035771558501627e137 or 1.046942143327783e126 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 19.8

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

    2. Taylor expanded around 0 19.6

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

    4. Applied associate-/l*11.7

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

    6. Applied *-un-lft-identity11.7

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

    7. Applied times-frac2.6

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

    8. Simplified2.6

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

    if -3.8035771558501627e137 < (- (* x y) (* (* z 9.0) t)) < 1.046942143327783e126

    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 add-cube-cbrt2.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t} \cdot \sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}\right) \cdot \sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}}{a \cdot 2}\]

    4. Applied associate-/l*2.0

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -3.8035771558501627 \cdot 10^{137} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.046942143327783 \cdot 10^{126}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t} \cdot \sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}{\frac{a \cdot 2}{\sqrt[3]{x \cdot y - \left(z \cdot 9\right) \cdot t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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 (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))