Average Error: 7.9 → 4.3
Time: 4.6s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot 2 \le -2.3404069236037078 \cdot 10^{68} \lor \neg \left(a \cdot 2 \le 2.4804157570983788 \cdot 10^{-48}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{y}{a} \cdot x\right) - 4.5 \cdot \left(\frac{t}{a} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x - 9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;a \cdot 2 \le -2.3404069236037078 \cdot 10^{68} \lor \neg \left(a \cdot 2 \le 2.4804157570983788 \cdot 10^{-48}\right):\\
\;\;\;\;0.5 \cdot \left(\frac{y}{a} \cdot x\right) - 4.5 \cdot \left(\frac{t}{a} \cdot z\right)\\

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

\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) (a * 2.0)) <= -2.340406923603708e+68) || !(((double) (a * 2.0)) <= 2.4804157570983788e-48))) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) (y / a)) * x)))) - ((double) (4.5 * ((double) (((double) (t / a)) * z))))));
	} else {
		VAR = ((double) (((double) (((double) (y * x)) - ((double) (9.0 * ((double) (t * z)))))) / ((double) (a * 2.0))));
	}
	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
Target5.7
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a 2.0) < -2.3404069236037078e68 or 2.4804157570983788e-48 < (* a 2.0)

    1. Initial program 11.5

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

    2. Simplified11.4

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

    4. Applied div-sub11.4

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

    5. Simplified9.1

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

    6. Simplified6.2

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

    7. Taylor expanded around 0 11.3

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

    8. Simplified5.7

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

    if -2.3404069236037078e68 < (* a 2.0) < 2.4804157570983788e-48

    1. Initial program 2.0

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

    2. Simplified2.0

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

    4. Applied div-sub2.0

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

    5. Simplified7.3

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

    6. Simplified11.4

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

    8. Applied associate-*r/7.3

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

    9. Applied associate-*r/2.0

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

    10. Applied sub-div2.0

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

    11. Simplified2.0

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \le -2.3404069236037078 \cdot 10^{68} \lor \neg \left(a \cdot 2 \le 2.4804157570983788 \cdot 10^{-48}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{y}{a} \cdot x\right) - 4.5 \cdot \left(\frac{t}{a} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x - 9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))