Average Error: 23.5 → 4.0
Time: 20.8s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.3265456538885105 \cdot 10^{+22} \lor \neg \left(z \leq 2.0874613586551125 \cdot 10^{+36}\right):\\ \;\;\;\;\begin{array}{l} t_1 := {\left(b - y\right)}^{2}\\ \mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_1}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{t_1}, \frac{t}{z}, \frac{a}{b - y}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt[3]{y + z \cdot \left(b - y\right)}\\ t_3 := \left(y + z \cdot b\right) - z \cdot y\\ \left(\frac{t}{\left(b + \frac{y}{z}\right) - y} + \frac{x \cdot \frac{y}{t_2 \cdot t_2}}{\sqrt[3]{t_3}}\right) - \frac{z \cdot a}{t_3} \end{array}\\ \end{array} \]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -3.3265456538885105 \cdot 10^{+22} \lor \neg \left(z \leq 2.0874613586551125 \cdot 10^{+36}\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 := \sqrt[3]{y + z \cdot \left(b - y\right)}\\
t_3 := \left(y + z \cdot b\right) - z \cdot y\\
\left(\frac{t}{\left(b + \frac{y}{z}\right) - y} + \frac{x \cdot \frac{y}{t_2 \cdot t_2}}{\sqrt[3]{t_3}}\right) - \frac{z \cdot a}{t_3}
\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 -3.3265456538885105e+22) (not (<= z 2.0874613586551125e+36)))
   (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 (cbrt (+ y (* z (- b y))))) (t_3 (- (+ y (* z b)) (* z y))))
     (-
      (+ (/ t (- (+ b (/ y z)) y)) (/ (* x (/ y (* t_2 t_2))) (cbrt t_3)))
      (/ (* z a) t_3)))))
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 <= -3.3265456538885105e+22) || !(z <= 2.0874613586551125e+36)) {
		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 = cbrt(y + (z * (b - y)));
		double t_3 = (y + (z * b)) - (z * y);
		tmp = ((t / ((b + (y / z)) - y)) + ((x * (y / (t_2 * t_2))) / cbrt(t_3))) - ((z * a) / t_3);
	}
	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

Original23.5
Target18.2
Herbie4.0
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.3265456538885105e22 or 2.0874613586551125e36 < z

    1. Initial program 41.0

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

    2. Taylor expanded in z around inf 21.7

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

    3. Simplified3.2

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

    if -3.3265456538885105e22 < z < 2.0874613586551125e36

    1. Initial program 9.1

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

    2. Taylor expanded in x around 0 9.1

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

    3. Applied associate-/l*_binary649.8

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

    4. Simplified9.8

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

    5. Taylor expanded in y around 0 9.7

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

    6. Applied add-cube-cbrt_binary6410.2

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

    7. Applied associate-/r*_binary6410.2

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

    8. Simplified4.7

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3265456538885105 \cdot 10^{+22} \lor \neg \left(z \leq 2.0874613586551125 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{\left(b + \frac{y}{z}\right) - y} + \frac{x \cdot \frac{y}{\sqrt[3]{y + z \cdot \left(b - y\right)} \cdot \sqrt[3]{y + z \cdot \left(b - y\right)}}}{\sqrt[3]{\left(y + z \cdot b\right) - z \cdot y}}\right) - \frac{z \cdot a}{\left(y + z \cdot b\right) - z \cdot y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))))