Average Error: 7.8 → 4.9
Time: 7.6s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -7.0026614161085481 \cdot 10^{204}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -3.0461796906836316 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 5.49127639575738348 \cdot 10^{-293}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.2137559508767057 \cdot 10^{246}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -7.0026614161085481 \cdot 10^{204}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -3.0461796906836316 \cdot 10^{-121}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 5.49127639575738348 \cdot 10^{-293}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.2137559508767057 \cdot 10^{246}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\

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

\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) (z * 9.0)) * t)) <= -7.002661416108548e+204)) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / a)))) - ((double) (4.5 * ((double) (t / ((double) (a / z))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (z * 9.0)) * t)) <= -3.0461796906836316e-121)) {
			VAR_1 = ((double) (((double) (0.5 * ((double) (x / ((double) (a / y)))))) - ((double) (4.5 * ((double) (((double) (t * z)) / a))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (z * 9.0)) * t)) <= 5.491276395757383e-293)) {
				VAR_2 = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / a)))) - ((double) (((double) (t * 4.5)) * ((double) (z / a))))));
			} else {
				double VAR_3;
				if ((((double) (((double) (z * 9.0)) * t)) <= 2.2137559508767057e+246)) {
					VAR_3 = ((double) (((double) (0.5 * ((double) (x / ((double) (a / y)))))) - ((double) (4.5 * ((double) (((double) (t * z)) / a))))));
				} else {
					VAR_3 = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / a)))) - ((double) (((double) (t * 4.5)) * ((double) (z / a))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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

Original7.8
Target5.6
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* z 9.0) t) < -7.0026614161085481e204

    1. Initial program 30.2

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

    2. Taylor expanded around 0 29.1

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

    4. Applied associate-/l*6.2

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

    if -7.0026614161085481e204 < (* (* z 9.0) t) < -3.0461796906836316e-121 or 5.49127639575738348e-293 < (* (* z 9.0) t) < 2.2137559508767057e246

    1. Initial program 3.9

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

    2. Taylor expanded around 0 3.9

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

    4. Applied associate-/l*4.4

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

    if -3.0461796906836316e-121 < (* (* z 9.0) t) < 5.49127639575738348e-293 or 2.2137559508767057e246 < (* (* z 9.0) t)

    1. Initial program 10.8

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

    2. Taylor expanded around 0 10.6

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

    4. Applied *-un-lft-identity10.6

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

    5. Applied times-frac5.6

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

    6. Applied associate-*r*5.6

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

    7. Simplified5.6

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -7.0026614161085481 \cdot 10^{204}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -3.0461796906836316 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 5.49127639575738348 \cdot 10^{-293}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 2.2137559508767057 \cdot 10^{246}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(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)))