Average Error: 21.9 → 7.4
Time: 15.0s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -4.310313169511819 \cdot 10^{+53} \lor \neg \left(z \leq 2.74295821737516 \cdot 10^{+27}\right):\\ \;\;\;\;\begin{array}{l} t_1 := {\left(b - y\right)}^{2}\\ \mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_1}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{t_1}, \frac{t}{z}, \frac{a}{b - y}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \left(y + z \cdot b\right) - z \cdot y\\ \left(\frac{z \cdot t}{t_2} + \frac{y \cdot x}{t_2}\right) - \frac{z \cdot a}{t_2} \end{array}\\ \end{array} \]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -4.310313169511819 \cdot 10^{+53} \lor \neg \left(z \leq 2.74295821737516 \cdot 10^{+27}\right):\\
\;\;\;\;\begin{array}{l}
t_1 := {\left(b - y\right)}^{2}\\
\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_1}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{t_1}, \frac{t}{z}, \frac{a}{b - y}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := \left(y + z \cdot b\right) - z \cdot y\\
\left(\frac{z \cdot t}{t_2} + \frac{y \cdot x}{t_2}\right) - \frac{z \cdot a}{t_2}
\end{array}\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.310313169511819e+53) (not (<= z 2.74295821737516e+27)))
   (let* ((t_1 (pow (- b y) 2.0)))
     (-
      (fma (/ y (- b y)) (/ x z) (fma (/ a t_1) (/ y z) (/ t (- b y))))
      (fma (/ y t_1) (/ t z) (/ a (- b y)))))
   (let* ((t_2 (- (+ y (* z b)) (* z y))))
     (- (+ (/ (* z t) t_2) (/ (* y x) t_2)) (/ (* z a) t_2)))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.310313169511819e+53) || !(z <= 2.74295821737516e+27)) {
		double t_1_1 = pow((b - y), 2.0);
		tmp = fma((y / (b - y)), (x / z), fma((a / t_1_1), (y / z), (t / (b - y)))) - fma((y / t_1_1), (t / z), (a / (b - y)));
	} else {
		double t_2 = (y + (z * b)) - (z * y);
		tmp = (((z * t) / t_2) + ((y * x) / t_2)) - ((z * a) / t_2);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original21.9
Target17.3
Herbie7.4
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.3103131695118189e53 or 2.74295821737516e27 < z

    1. Initial program 39.1

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

    2. Simplified39.1

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

    3. Taylor expanded in z around inf 22.3

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

    4. Simplified5.3

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

    if -4.3103131695118189e53 < z < 2.74295821737516e27

    1. Initial program 8.8

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

    2. Simplified8.8

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

    3. Taylor expanded in x around 0 9.0

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.310313169511819 \cdot 10^{+53} \lor \neg \left(z \leq 2.74295821737516 \cdot 10^{+27}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z \cdot t}{\left(y + z \cdot b\right) - z \cdot y} + \frac{y \cdot x}{\left(y + z \cdot b\right) - z \cdot y}\right) - \frac{z \cdot a}{\left(y + z \cdot b\right) - z \cdot y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))