Average Error: 7.8 → 4.8
Time: 8.7s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[[z, t]=\mathsf{sort}([z, t])\]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq -8.178025020792348 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{\frac{a}{\frac{y}{2}}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.2055081063493506 \cdot 10^{-308}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.1375185471495733 \cdot 10^{+49}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{4.5 \cdot \left(z \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \end{array}\]
\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 \leq -\infty:\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq -8.178025020792348 \cdot 10^{-85}:\\
\;\;\;\;\frac{x}{\frac{a}{\frac{y}{2}}} - 4.5 \cdot \frac{z \cdot t}{a}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.2055081063493506 \cdot 10^{-308}:\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.1375185471495733 \cdot 10^{+49}:\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{4.5 \cdot \left(z \cdot t\right)}{a}\\

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

\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 (<= (* (* z 9.0) t) (- INFINITY))
   (- (/ (* x y) (* a 2.0)) (* (* t (/ z a)) 4.5))
   (if (<= (* (* z 9.0) t) -8.178025020792348e-85)
     (- (/ x (/ a (/ y 2.0))) (* 4.5 (/ (* z t) a)))
     (if (<= (* (* z 9.0) t) 1.2055081063493506e-308)
       (- (/ (* x y) (* a 2.0)) (* 4.5 (/ t (/ a z))))
       (if (<= (* (* z 9.0) t) 1.1375185471495733e+49)
         (- (/ (* x y) (* a 2.0)) (/ (* 4.5 (* z t)) a))
         (- (/ (* x y) (* a 2.0)) (* (* t (/ z a)) 4.5)))))))
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 (((z * 9.0) * t) <= -((double) INFINITY)) {
		tmp = ((x * y) / (a * 2.0)) - ((t * (z / a)) * 4.5);
	} else if (((z * 9.0) * t) <= -8.178025020792348e-85) {
		tmp = (x / (a / (y / 2.0))) - (4.5 * ((z * t) / a));
	} else if (((z * 9.0) * t) <= 1.2055081063493506e-308) {
		tmp = ((x * y) / (a * 2.0)) - (4.5 * (t / (a / z)));
	} else if (((z * 9.0) * t) <= 1.1375185471495733e+49) {
		tmp = ((x * y) / (a * 2.0)) - ((4.5 * (z * t)) / a);
	} else {
		tmp = ((x * y) / (a * 2.0)) - ((t * (z / a)) * 4.5);
	}
	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

Original7.8
Target5.4
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (*.f64 (*.f64 z 9) t) < -inf.0 or 1.1375185471495733e49 < (*.f64 (*.f64 z 9) t)

    1. Initial program 22.1

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

    3. Applied div-sub_binary6422.1

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

    4. Simplified21.6

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

    6. Applied *-un-lft-identity_binary6421.6

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

    7. Applied times-frac_binary648.5

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

    8. Simplified8.5

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

    if -inf.0 < (*.f64 (*.f64 z 9) t) < -8.1780250207923483e-85

    1. Initial program 3.6

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

    3. Applied div-sub_binary643.6

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

    4. Simplified3.6

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

    6. Applied associate-/l*_binary643.1

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

    7. Simplified3.0

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

    if -8.1780250207923483e-85 < (*.f64 (*.f64 z 9) t) < 1.205508106349351e-308

    1. Initial program 4.6

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

    3. Applied div-sub_binary644.6

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

    4. Simplified4.7

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

    6. Applied associate-/l*_binary644.5

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

    if 1.205508106349351e-308 < (*.f64 (*.f64 z 9) t) < 1.1375185471495733e49

    1. Initial program 4.1

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

    3. Applied div-sub_binary644.1

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

    4. Simplified4.1

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

    6. Applied associate-*l/_binary644.1

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{\left(t \cdot z\right) \cdot 4.5}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq -8.178025020792348 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{\frac{a}{\frac{y}{2}}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.2055081063493506 \cdot 10^{-308}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 1.1375185471495733 \cdot 10^{+49}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{4.5 \cdot \left(z \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \left(t \cdot \frac{z}{a}\right) \cdot 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2021147 
(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)))