Average Error: 7.9 → 5.6
Time: 3.9s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -4.51662160705808014 \cdot 10^{286}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -5.59555531709714142 \cdot 10^{-173}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 7.7104992084836218 \cdot 10^{-145}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -4.51662160705808014 \cdot 10^{286}:\\
\;\;\;\;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 -5.59555531709714142 \cdot 10^{-173}:\\
\;\;\;\;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 7.7104992084836218 \cdot 10^{-145}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot 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)) <= -4.51662160705808e+286)) {
		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)) <= -5.5955553170971414e-173)) {
			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)) <= 7.710499208483622e-145)) {
				VAR_2 = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / a)))) - ((double) (4.5 * ((double) (t / ((double) (a / z))))))));
			} else {
				VAR_2 = ((double) (((double) (0.5 * ((double) (x * ((double) (y / a)))))) - ((double) (4.5 * ((double) (((double) (t * z)) / a))))));
			}
			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

Original7.9
Target5.5
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* z 9.0) t) < -4.51662160705808e+286 or -5.5955553170971414e-173 < (* (* z 9.0) t) < 7.710499208483622e-145

    1. Initial program 10.3

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

    2. Taylor expanded around 0 10.2

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

    4. Applied associate-/l*5.0

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

    if -4.51662160705808e+286 < (* (* z 9.0) t) < -5.5955553170971414e-173

    1. Initial program 4.0

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

    2. Taylor expanded around 0 4.0

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

    4. Applied associate-/l*3.9

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

    if 7.710499208483622e-145 < (* (* z 9.0) t)

    1. Initial program 9.0

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

    2. Taylor expanded around 0 8.8

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

    4. Applied *-un-lft-identity8.8

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

    5. Applied times-frac7.8

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

    6. Simplified7.8

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -4.51662160705808014 \cdot 10^{286}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -5.59555531709714142 \cdot 10^{-173}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 7.7104992084836218 \cdot 10^{-145}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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)))