Average Error: 23.5 → 8.7
Time: 20.4s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\left(\frac{a}{y - b} + \frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right)\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -4.5537789761596633 \cdot 10^{-256}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\left(\frac{a}{y - b} + \frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right)\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 3.8123447295996716 \cdot 10^{+289}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\
\;\;\;\;\left(\frac{a}{y - b} + \frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right)\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -4.5537789761596633 \cdot 10^{-256}:\\
\;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\
\;\;\;\;\left(\frac{a}{y - b} + \frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right)\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 3.8123447295996716 \cdot 10^{+289}:\\
\;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\
\;\;\;\;\frac{x}{1 - z}\\

\mathbf{else}:\\
\;\;\;\;\frac{t - a}{b - y}\\

\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 (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) (- INFINITY))
   (-
    (+ (/ a (- y b)) (* (/ y z) (- (/ a (pow (- y b) 2.0)) (/ x (- y b)))))
    (+ (/ t (- y b)) (/ (* y t) (* z (pow (- y b) 2.0)))))
   (if (<=
        (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
        -4.5537789761596633e-256)
     (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
     (if (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) 0.0)
       (-
        (+ (/ a (- y b)) (* (/ y z) (- (/ a (pow (- y b) 2.0)) (/ x (- y b)))))
        (+ (/ t (- y b)) (/ (* y t) (* z (pow (- y b) 2.0)))))
       (if (<=
            (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
            3.8123447295996716e+289)
         (/ (- (+ (* x y) (* z t)) (* z a)) (+ y (* z (- b y))))
         (if (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) INFINITY)
           (/ x (- 1.0 z))
           (/ (- t a) (- b y))))))))
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 ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -((double) INFINITY)) {
		tmp = ((a / (y - b)) + ((y / z) * ((a / pow((y - b), 2.0)) - (x / (y - b))))) - ((t / (y - b)) + ((y * t) / (z * pow((y - b), 2.0))));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -4.5537789761596633e-256) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 0.0) {
		tmp = ((a / (y - b)) + ((y / z) * ((a / pow((y - b), 2.0)) - (x / (y - b))))) - ((t / (y - b)) + ((y * t) / (z * pow((y - b), 2.0))));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 3.8123447295996716e+289) {
		tmp = (((x * y) + (z * t)) - (z * a)) / (y + (z * (b - y)));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= ((double) INFINITY)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / (b - y);
	}
	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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original23.5
Target18.2
Herbie8.7
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 5 regimes

  2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or -4.55377897615966332e-256 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

    1. Initial program 51.9

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

    2. Taylor expanded around 0 52.5

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

    3. Simplified51.9

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

    4. Taylor expanded around -inf 28.2

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

    5. Simplified20.9

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

    if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.55377897615966332e-256

    1. Initial program 0.3

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

    if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 3.8123447295996716e289

    1. Initial program 0.3

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

    3. Applied sub-neg_binary64_191670.3

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

    4. Applied distribute-rgt-in_binary64_191240.3

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

    5. Applied associate-+r+_binary64_191060.3

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

    if 3.8123447295996716e289 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

    1. Initial program 59.6

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

    2. Taylor expanded around inf 31.2

      \[\leadsto \color{blue}{\frac{x}{1 - z}}\]

    if +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

    1. Initial program 64.0

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

    2. Taylor expanded around inf 16.3

      \[\leadsto \color{blue}{\frac{t - a}{b - y}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\left(\frac{a}{y - b} + \frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right)\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -4.5537789761596633 \cdot 10^{-256}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\left(\frac{a}{y - b} + \frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right)\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 3.8123447295996716 \cdot 10^{+289}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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)))))