Average Error: 7.7 → 1.2
Time: 16.2s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[[z, t]=\mathsf{sort}([z, t])\]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4.7642198436170626 \cdot 10^{+296} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.430354221040605 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{x}{\frac{a}{\frac{y}{2}}} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(z \cdot t\right) \cdot \frac{4.5}{a}\\ \end{array}\]
\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 \leq -4.7642198436170626 \cdot 10^{+296} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.430354221040605 \cdot 10^{+158}\right):\\
\;\;\;\;\frac{x}{\frac{a}{\frac{y}{2}}} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(z \cdot t\right) \cdot \frac{4.5}{a}\\

\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)) -4.7642198436170626e+296)
         (not (<= (- (* x y) (* (* z 9.0) t)) 2.430354221040605e+158)))
   (- (/ x (/ a (/ y 2.0))) (* (* t (/ z a)) 4.5))
   (- (/ (* x y) (* a 2.0)) (* (* z t) (/ 4.5 a)))))
double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((x * y) - ((z * 9.0) * t)) <= -4.7642198436170626e+296) || !(((x * y) - ((z * 9.0) * t)) <= 2.430354221040605e+158)) {
		tmp = (x / (a / (y / 2.0))) - ((t * (z / a)) * 4.5);
	} else {
		tmp = ((x * y) / (a * 2.0)) - ((z * t) * (4.5 / a));
	}
	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

Original7.7
Target5.3
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -4.76421984361706262e296 or 2.4303542210406052e158 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

    1. Initial program 31.0

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

    3. Applied div-sub_binary64_1611031.0

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

    4. Simplified30.7

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{t \cdot z}{a} \cdot 4.5}\]
    5. Using strategy rm

    6. Applied associate-/l*_binary64_1605017.3

      \[\leadsto \color{blue}{\frac{x}{\frac{a \cdot 2}{y}}} - \frac{t \cdot z}{a} \cdot 4.5\]

    7. Simplified17.2

      \[\leadsto \frac{x}{\color{blue}{\frac{a}{\frac{y}{2}}}} - \frac{t \cdot z}{a} \cdot 4.5\]
    8. Using strategy rm

    9. Applied *-un-lft-identity_binary64_1610517.2

      \[\leadsto \frac{x}{\frac{a}{\frac{y}{2}}} - \frac{t \cdot z}{\color{blue}{1 \cdot a}} \cdot 4.5\]

    10. Applied times-frac_binary64_161111.8

      \[\leadsto \frac{x}{\frac{a}{\frac{y}{2}}} - \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)} \cdot 4.5\]

    if -4.76421984361706262e296 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 2.4303542210406052e158

    1. Initial program 1.0

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

    3. Applied div-sub_binary64_161101.0

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

    4. Simplified1.0

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{t \cdot z}{a} \cdot 4.5}\]
    5. Using strategy rm

    6. Applied div-inv_binary64_161021.0

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{a}\right)} \cdot 4.5\]

    7. Applied associate-*l*_binary64_160461.1

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\left(t \cdot z\right) \cdot \left(\frac{1}{a} \cdot 4.5\right)}\]

    8. Simplified1.0

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4.7642198436170626 \cdot 10^{+296} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.430354221040605 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{x}{\frac{a}{\frac{y}{2}}} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(z \cdot t\right) \cdot \frac{4.5}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))