Average Error: 7.1 → 0.8
Time: 1.4m
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.16976563187768801 \cdot 10^{285} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.7823364456778995 \cdot 10^{246}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{x \cdot 1}{1}, \frac{y}{a}, -\frac{z}{a} \cdot \left(4.5 \cdot \frac{t \cdot 1}{1}\right)\right) + \mathsf{fma}\left(-\frac{z}{a}, 4.5 \cdot \frac{t \cdot 1}{1}, \frac{z}{a} \cdot \left(4.5 \cdot \frac{t \cdot 1}{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(-x \cdot y\right) - 4.5 \cdot \left(-t \cdot z\right)}{-a}\\ \end{array}\]
\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 \le -5.16976563187768801 \cdot 10^{285} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.7823364456778995 \cdot 10^{246}\right):\\
\;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{x \cdot 1}{1}, \frac{y}{a}, -\frac{z}{a} \cdot \left(4.5 \cdot \frac{t \cdot 1}{1}\right)\right) + \mathsf{fma}\left(-\frac{z}{a}, 4.5 \cdot \frac{t \cdot 1}{1}, \frac{z}{a} \cdot \left(4.5 \cdot \frac{t \cdot 1}{1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5 \cdot \left(-x \cdot y\right) - 4.5 \cdot \left(-t \cdot z\right)}{-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) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= -5.169765631877688e+285) || !(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 1.7823364456778995e+246))) {
		VAR = ((double) (((double) fma(((double) (0.5 * ((double) (((double) (x * 1.0)) / 1.0)))), ((double) (y / a)), ((double) -(((double) (((double) (z / a)) * ((double) (4.5 * ((double) (((double) (t * 1.0)) / 1.0)))))))))) + ((double) fma(((double) -(((double) (z / a)))), ((double) (4.5 * ((double) (((double) (t * 1.0)) / 1.0)))), ((double) (((double) (z / a)) * ((double) (4.5 * ((double) (((double) (t * 1.0)) / 1.0))))))))));
	} else {
		VAR = ((double) (((double) (((double) (0.5 * ((double) -(((double) (x * y)))))) - ((double) (4.5 * ((double) -(((double) (t * z)))))))) / ((double) -(a))));
	}
	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

Original7.1
Target5.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -5.169765631877688e+285 or 1.7823364456778995e+246 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 42.2

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

    2. Taylor expanded around 0 41.9

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

    4. Applied *-un-lft-identity41.9

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

    5. Applied associate-*r*41.9

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

    6. Applied associate-/l*22.9

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x \cdot 1}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity22.9

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

    9. Applied *-un-lft-identity22.9

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

    10. Applied associate-*r*22.9

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

    11. Applied times-frac0.4

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

    12. Applied associate-*r*0.5

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

    13. Applied clear-num0.6

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

    14. Applied associate-/r/0.5

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

    15. Applied associate-*r*0.5

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

    16. Applied prod-diff0.5

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

    if -5.169765631877688e+285 < (- (* x y) (* (* z 9.0) t)) < 1.7823364456778995e+246

    1. Initial program 0.9

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

    2. Taylor expanded around 0 0.9

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

    4. Applied frac-2neg0.9

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

    5. Applied associate-*r/0.9

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

    6. Applied frac-2neg0.9

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

    7. Applied associate-*r/0.9

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

    8. Applied sub-div0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.16976563187768801 \cdot 10^{285} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.7823364456778995 \cdot 10^{246}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{x \cdot 1}{1}, \frac{y}{a}, -\frac{z}{a} \cdot \left(4.5 \cdot \frac{t \cdot 1}{1}\right)\right) + \mathsf{fma}\left(-\frac{z}{a}, 4.5 \cdot \frac{t \cdot 1}{1}, \frac{z}{a} \cdot \left(4.5 \cdot \frac{t \cdot 1}{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(-x \cdot y\right) - 4.5 \cdot \left(-t \cdot z\right)}{-a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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)))