Average Error: 7.6 → 0.9
Time: 6.2s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -9.450001500785823 \cdot 10^{+234} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 1.2657093729560723 \cdot 10^{+272}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]
\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 \leq -9.450001500785823 \cdot 10^{+234} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 1.2657093729560723 \cdot 10^{+272}\right):\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\

\end{array}
(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- (* x y) (* (* z 9.0) t)) -9.450001500785823e+234)
         (not (<= (- (* x y) (* (* z 9.0) t)) 1.2657093729560723e+272)))
   (- (* 0.5 (* x (/ y a))) (* 4.5 (* z (/ t a))))
   (* (/ 1.0 a) (/ (- (* x y) (* (* z 9.0) t)) 2.0))))
double code(double x, double y, double z, double t, double a) {
	return (((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 tmp;
	if (((((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= -9.450001500785823e+234) || !(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 1.2657093729560723e+272))) {
		tmp = ((double) (((double) (0.5 * ((double) (x * (y / a))))) - ((double) (4.5 * ((double) (z * (t / a)))))));
	} else {
		tmp = ((double) ((1.0 / a) * (((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) / 2.0)));
	}
	return tmp;
}

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

Original7.6
Target5.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9.0) t)) < -9.4500015007858227e234 or 1.2657093729560723e272 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9.0) t))

    1. Initial program 38.8

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

    2. Taylor expanded around 0 38.3

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

    3. Simplified38.3

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

    5. Applied *-un-lft-identity_binary6438.3

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

    6. Applied times-frac_binary6421.2

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

    7. Simplified21.2

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

    9. Applied *-un-lft-identity_binary6421.2

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

    10. Applied times-frac_binary640.6

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

    11. Simplified0.6

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

    if -9.4500015007858227e234 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9.0) t)) < 1.2657093729560723e272

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity_binary640.9

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]

    4. Applied times-frac_binary640.9

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -9.450001500785823 \cdot 10^{+234} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 1.2657093729560723 \cdot 10^{+272}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))