Average Error: 7.6 → 5.2
Time: 5.4s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.774026145451886 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -0:\\ \;\;\;\;\left(x \cdot y - t \cdot \left(z \cdot 9\right)\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 1.002744172180215 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 1.514157687347536 \cdot 10^{+179}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - t \cdot \left(4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \leq -4.774026145451886 \cdot 10^{+109}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\

\mathbf{elif}\;x \cdot y \leq -0:\\
\;\;\;\;\left(x \cdot y - t \cdot \left(z \cdot 9\right)\right) \cdot \frac{0.5}{a}\\

\mathbf{elif}\;x \cdot y \leq 1.002744172180215 \cdot 10^{-61}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\

\mathbf{elif}\;x \cdot y \leq 1.514157687347536 \cdot 10^{+179}:\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - t \cdot \left(4.5 \cdot \frac{z}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{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 (<= (* x y) -4.774026145451886e+109)
   (- (* 0.5 (/ x (/ a y))) (* 4.5 (/ (* z t) a)))
   (if (<= (* x y) -0.0)
     (* (- (* x y) (* t (* z 9.0))) (/ 0.5 a))
     (if (<= (* x y) 1.002744172180215e-61)
       (- (* 0.5 (/ (* x y) a)) (* 4.5 (* z (/ t a))))
       (if (<= (* x y) 1.514157687347536e+179)
         (- (/ (* x y) (* a 2.0)) (* t (* 4.5 (/ z a))))
         (- (* 0.5 (/ x (/ a y))) (* 4.5 (/ (* z t) 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) <= -4.774026145451886e+109) {
		tmp = (0.5 * (x / (a / y))) - (4.5 * ((z * t) / a));
	} else if ((x * y) <= -0.0) {
		tmp = ((x * y) - (t * (z * 9.0))) * (0.5 / a);
	} else if ((x * y) <= 1.002744172180215e-61) {
		tmp = (0.5 * ((x * y) / a)) - (4.5 * (z * (t / a)));
	} else if ((x * y) <= 1.514157687347536e+179) {
		tmp = ((x * y) / (a * 2.0)) - (t * (4.5 * (z / a)));
	} else {
		tmp = (0.5 * (x / (a / y))) - (4.5 * ((z * t) / 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

Original7.6
Target5.5
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (*.f64 x y) < -4.77402614545188629e109 or 1.5141576873475359e179 < (*.f64 x y)

    1. Initial program 22.7

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

    3. Applied clear-num_binary6422.8

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

    4. Simplified22.8

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

    5. Taylor expanded around inf 22.7

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

    6. Simplified22.7

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

    8. Applied associate-/l*_binary648.2

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

    if -4.77402614545188629e109 < (*.f64 x y) < -0.0

    1. Initial program 4.0

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

    3. Applied div-inv_binary644.1

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

    4. Simplified4.1

      \[\leadsto \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \color{blue}{\frac{0.5}{a}}\]

    if -0.0 < (*.f64 x y) < 1.00274417218021508e-61

    1. Initial program 3.5

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

    3. Applied clear-num_binary643.9

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

    4. Simplified3.9

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

    5. Taylor expanded around inf 3.5

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

    6. Simplified3.5

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

    8. Applied *-un-lft-identity_binary643.5

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

    9. Applied times-frac_binary646.2

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

    10. Simplified6.2

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

    if 1.00274417218021508e-61 < (*.f64 x y) < 1.5141576873475359e179

    1. Initial program 3.4

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

    3. Applied div-sub_binary643.4

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

    4. Simplified3.4

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.774026145451886 \cdot 10^{+109}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -0:\\ \;\;\;\;\left(x \cdot y - t \cdot \left(z \cdot 9\right)\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 1.002744172180215 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 1.514157687347536 \cdot 10^{+179}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - t \cdot \left(4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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)))