Average Error: 7.2 → 0.7
Time: 8.9s
Precision: binary64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -1.5851593702228302 \cdot 10^{+295}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t_1 \leq 1.5438915637834441 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x \cdot y}{a}, \frac{z \cdot t}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(y \cdot \frac{1}{a}\right)\right) - 4.5 \cdot \frac{z}{\frac{a}{t}}\\ \end{array} \]
\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 -1.5851593702228302 \cdot 10^{+295}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\

\mathbf{elif}\;t_1 \leq 1.5438915637834441 \cdot 10^{+283}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{x \cdot y}{a}, \frac{z \cdot t}{a} \cdot -4.5\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \left(y \cdot \frac{1}{a}\right)\right) - 4.5 \cdot \frac{z}{\frac{a}{t}}\\


\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 -1.5851593702228302e+295)
     (- (* 0.5 (* x (/ y a))) (* 4.5 (* z (/ t a))))
     (if (<= t_1 1.5438915637834441e+283)
       (fma 0.5 (/ (* x y) a) (* (/ (* z t) a) -4.5))
       (- (* 0.5 (* x (* y (/ 1.0 a)))) (* 4.5 (/ z (/ a t))))))))
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 <= -1.5851593702228302e+295) {
		tmp = (0.5 * (x * (y / a))) - (4.5 * (z * (t / a)));
	} else if (t_1 <= 1.5438915637834441e+283) {
		tmp = fma(0.5, ((x * y) / a), (((z * t) / a) * -4.5));
	} else {
		tmp = (0.5 * (x * (y * (1.0 / a)))) - (4.5 * (z / (a / t)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.2
Target5.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -1.5851593702228302e295

    1. Initial program 57.0

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

    2. Simplified56.8

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

    3. Taylor expanded in z around 0 56.6

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

    4. Applied egg-rr32.6

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

    5. Applied egg-rr0.3

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

    if -1.5851593702228302e295 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.54389156378344414e283

    1. Initial program 0.7

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

    2. Simplified0.8

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

    3. Taylor expanded in z around 0 0.7

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

    4. Applied egg-rr0.7

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

    if 1.54389156378344414e283 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

    1. Initial program 49.7

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

    2. Simplified49.6

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

    3. Taylor expanded in z around 0 49.2

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

    4. Applied egg-rr26.5

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

    5. Applied egg-rr26.5

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

    6. Applied egg-rr0.6

      \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(y \cdot \frac{1}{a}\right)\right)} - 4.5 \cdot \frac{z}{\frac{a}{t}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1.5851593702228302 \cdot 10^{+295}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 1.5438915637834441 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x \cdot y}{a}, \frac{z \cdot t}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(y \cdot \frac{1}{a}\right)\right) - 4.5 \cdot \frac{z}{\frac{a}{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))