Average Error: 7.6 → 4.5
Time: 5.7s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5.368315857521777 \cdot 10^{+221}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.5696795690357024 \cdot 10^{+293}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(z \cdot 4.5\right) \cdot \frac{t}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5.368315857521777 \cdot 10^{+221}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\

\mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.5696795690357024 \cdot 10^{+293}:\\
\;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(z \cdot 4.5\right) \cdot \frac{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) (* (* z 9.0) t)) -5.368315857521777e+221)
   (- (* 0.5 (/ x (/ a y))) (* 4.5 (/ (* z t) a)))
   (if (<= (- (* x y) (* (* z 9.0) t)) 2.5696795690357024e+293)
     (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))
     (- (* 0.5 (/ (* x y) a)) (* (* z 4.5) (/ 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) - ((z * 9.0) * t)) <= -5.368315857521777e+221) {
		tmp = (0.5 * (x / (a / y))) - (4.5 * ((z * t) / a));
	} else if (((x * y) - ((z * 9.0) * t)) <= 2.5696795690357024e+293) {
		tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	} else {
		tmp = (0.5 * ((x * y) / a)) - ((z * 4.5) * (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
Target5.6
Herbie4.5
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -5.368315857521777e221

    1. Initial program 33.5

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

    3. Applied sub-neg_binary64_2155433.5

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

    4. Simplified33.5

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

    5. Taylor expanded around 0 33.4

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

    6. Simplified33.4

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

    8. Applied associate-/l*_binary64_2150618.0

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

    if -5.368315857521777e221 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 2.56967956903570236e293

    1. Initial program 0.9

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

    if 2.56967956903570236e293 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

    1. Initial program 56.3

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

    3. Applied sub-neg_binary64_2155456.3

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

    4. Simplified55.8

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

    5. Taylor expanded around 0 55.1

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

    6. Simplified55.1

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

    8. Applied *-un-lft-identity_binary64_2156155.1

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

    9. Applied times-frac_binary64_2156731.3

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

    10. Applied associate-*r*_binary64_2150131.7

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

    11. Simplified31.7

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

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5.368315857521777 \cdot 10^{+221}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.5696795690357024 \cdot 10^{+293}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(z \cdot 4.5\right) \cdot \frac{t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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)))