Average Error: 7.8 → 3.2
Time: 9.0s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \[z, t] = \mathsf{sort}([z, t]) \\]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -5.167525457868286 \cdot 10^{+204}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right) - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t_1 \leq 1.6768450494954835 \cdot 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\
\mathbf{if}\;t_1 \leq -5.167525457868286 \cdot 10^{+204}:\\
\;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right) - 4.5 \cdot \frac{z \cdot t}{a}\\

\mathbf{elif}\;t_1 \leq 1.6768450494954835 \cdot 10^{+217}:\\
\;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -9, x \cdot y\right)}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\


\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
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 -5.167525457868286e+204)
     (- (* 0.5 (* y (/ x a))) (* 4.5 (/ (* z t) a)))
     (if (<= t_1 1.6768450494954835e+217)
       (/ (fma t (* z -9.0) (* x y)) (* a 2.0))
       (-
        (* 0.5 (/ (* x (/ y (* (cbrt a) (cbrt a)))) (cbrt a)))
        (* 4.5 (/ t (/ a z))))))))
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 t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -5.167525457868286e+204) {
		tmp = (0.5 * (y * (x / a))) - (4.5 * ((z * t) / a));
	} else if (t_1 <= 1.6768450494954835e+217) {
		tmp = fma(t, (z * -9.0), (x * y)) / (a * 2.0);
	} else {
		tmp = (0.5 * ((x * (y / (cbrt(a) * cbrt(a)))) / cbrt(a))) - (4.5 * (t / (a / z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.8
Target5.7
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -5.1675254578682864e204

    1. Initial program 28.4

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

    2. Taylor expanded in x around 0 28.0

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

    3. Applied *-un-lft-identity_binary6428.0

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

    4. Applied times-frac_binary6415.5

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

    5. Simplified15.5

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

    if -5.1675254578682864e204 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.67684504949548346e217

    1. Initial program 1.0

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

    2. Taylor expanded in x around 0 1.0

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

    3. Simplified1.0

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

    if 1.67684504949548346e217 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

    1. Initial program 32.1

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

    2. Taylor expanded in x around 0 31.7

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

    3. Applied add-cube-cbrt_binary6431.9

      \[\leadsto 0.5 \cdot \frac{y \cdot x}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a} \]

    4. Applied associate-/r*_binary6431.9

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{y \cdot x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a} \]

    5. Simplified20.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}}{\sqrt[3]{a}} - 4.5 \cdot \frac{t \cdot z}{a} \]

    6. Applied associate-/l*_binary645.9

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5.167525457868286 \cdot 10^{+204}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right) - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 1.6768450494954835 \cdot 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))