Average Error: 23.1 → 9.0
Time: 41.8s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\left(\left(\frac{a}{y - b} + \frac{y \cdot a}{z \cdot {\left(y - b\right)}^{2}}\right) + \frac{a \cdot \left(y \cdot y\right)}{\left(z \cdot z\right) \cdot {\left(y - b\right)}^{3}}\right) - \left(\frac{y}{z} \cdot \left(\frac{x}{y - b} + \frac{t}{{\left(y - b\right)}^{2}}\right) + \left(\frac{t}{y - b} + \left(\frac{y}{z} \cdot \frac{y}{z}\right) \cdot \left(\frac{x}{{\left(y - b\right)}^{2}} + \frac{t}{{\left(y - b\right)}^{3}}\right)\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1.1472190365846532 \cdot 10^{-285}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\left(\frac{t}{b - y} + \left(\frac{y \cdot \frac{x}{z}}{b - y} + \frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}}\right)\right) - \left(\frac{t}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}} + \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2.1850262770065685 \cdot 10^{+286}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array}\]
\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(\left(\frac{a}{y - b} + \frac{y \cdot a}{z \cdot {\left(y - b\right)}^{2}}\right) + \frac{a \cdot \left(y \cdot y\right)}{\left(z \cdot z\right) \cdot {\left(y - b\right)}^{3}}\right) - \left(\frac{y}{z} \cdot \left(\frac{x}{y - b} + \frac{t}{{\left(y - b\right)}^{2}}\right) + \left(\frac{t}{y - b} + \left(\frac{y}{z} \cdot \frac{y}{z}\right) \cdot \left(\frac{x}{{\left(y - b\right)}^{2}} + \frac{t}{{\left(y - b\right)}^{3}}\right)\right)\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1.1472190365846532 \cdot 10^{-285}:\\
\;\;\;\;\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 0:\\
\;\;\;\;\left(\frac{t}{b - y} + \left(\frac{y \cdot \frac{x}{z}}{b - y} + \frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}}\right)\right) - \left(\frac{t}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}} + \frac{a}{b - y}\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2.1850262770065685 \cdot 10^{+286}:\\
\;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\

\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 a) (* z (pow (- y b) 2.0))))
     (/ (* a (* y y)) (* (* z z) (pow (- y b) 3.0))))
    (+
     (* (/ y z) (+ (/ x (- y b)) (/ t (pow (- y b) 2.0))))
     (+
      (/ t (- y b))
      (*
       (* (/ y z) (/ y z))
       (+ (/ x (pow (- y b) 2.0)) (/ t (pow (- y b) 3.0)))))))
   (if (<=
        (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
        -1.1472190365846532e-285)
     (/ (- (+ (* x y) (* z t)) (* z a)) (+ y (* z (- b y))))
     (if (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) 0.0)
       (-
        (+
         (/ t (- b y))
         (+ (/ (* y (/ x z)) (- b y)) (/ (* y a) (* z (pow (- b y) 2.0)))))
        (+ (* (/ t z) (/ y (pow (- b y) 2.0))) (/ a (- b y))))
       (if (<=
            (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
            2.1850262770065685e+286)
         (/ (- (+ (* x y) (* z t)) (* z a)) (+ y (* z (- b y))))
         (/ (- 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 * a) / (z * pow((y - b), 2.0)))) + ((a * (y * y)) / ((z * z) * pow((y - b), 3.0)))) - (((y / z) * ((x / (y - b)) + (t / pow((y - b), 2.0)))) + ((t / (y - b)) + (((y / z) * (y / z)) * ((x / pow((y - b), 2.0)) + (t / pow((y - b), 3.0))))));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -1.1472190365846532e-285) {
		tmp = (((x * y) + (z * t)) - (z * a)) / (y + (z * (b - y)));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 0.0) {
		tmp = ((t / (b - y)) + (((y * (x / z)) / (b - y)) + ((y * a) / (z * pow((b - y), 2.0))))) - (((t / z) * (y / pow((b - y), 2.0))) + (a / (b - y)));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 2.1850262770065685e+286) {
		tmp = (((x * y) + (z * t)) - (z * a)) / (y + (z * (b - y)));
	} 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

Original23.1
Target17.8
Herbie9.0
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 4 regimes

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

    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 42.6

      \[\leadsto \color{blue}{\left(\frac{a \cdot {y}^{2}}{{\left(y - b\right)}^{3} \cdot {z}^{2}} + \left(\frac{a}{y - b} + \frac{a \cdot y}{{\left(y - b\right)}^{2} \cdot z}\right)\right) - \left(\frac{x \cdot y}{\left(y - b\right) \cdot z} + \left(\frac{t \cdot y}{{\left(y - b\right)}^{2} \cdot z} + \left(\frac{t}{y - b} + \left(\frac{x \cdot {y}^{2}}{{\left(y - b\right)}^{2} \cdot {z}^{2}} + \frac{t \cdot {y}^{2}}{{\left(y - b\right)}^{3} \cdot {z}^{2}}\right)\right)\right)\right)}\]

    3. Simplified42.1

      \[\leadsto \color{blue}{\left(\left(\frac{a}{y - b} + \frac{a \cdot y}{z \cdot {\left(y - b\right)}^{2}}\right) + \frac{a \cdot \left(y \cdot y\right)}{\left(z \cdot z\right) \cdot {\left(y - b\right)}^{3}}\right) - \left(\frac{y}{z} \cdot \left(\frac{x}{y - b} + \frac{t}{{\left(y - b\right)}^{2}}\right) + \left(\frac{t}{y - b} + \left(\frac{y}{z} \cdot \frac{y}{z}\right) \cdot \left(\frac{x}{{\left(y - b\right)}^{2}} + \frac{t}{{\left(y - b\right)}^{3}}\right)\right)\right)}\]

    if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.1472190365846532e-285 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.18502627700656848e286

    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_171210.3

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

    4. Applied distribute-rgt-in_binary64_170780.3

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

    5. Applied associate-+r+_binary64_170600.3

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

    if -1.1472190365846532e-285 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

    1. Initial program 45.7

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

    2. Taylor expanded around inf 19.8

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

    3. Simplified19.8

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

    5. Applied associate-/r*_binary64_170729.8

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

    6. Simplified9.6

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

    8. Applied times-frac_binary64_171346.8

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

    if 2.18502627700656848e286 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

    1. Initial program 62.2

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

    2. Taylor expanded around inf 22.6

      \[\leadsto \color{blue}{\frac{t - a}{b - y}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\left(\left(\frac{a}{y - b} + \frac{y \cdot a}{z \cdot {\left(y - b\right)}^{2}}\right) + \frac{a \cdot \left(y \cdot y\right)}{\left(z \cdot z\right) \cdot {\left(y - b\right)}^{3}}\right) - \left(\frac{y}{z} \cdot \left(\frac{x}{y - b} + \frac{t}{{\left(y - b\right)}^{2}}\right) + \left(\frac{t}{y - b} + \left(\frac{y}{z} \cdot \frac{y}{z}\right) \cdot \left(\frac{x}{{\left(y - b\right)}^{2}} + \frac{t}{{\left(y - b\right)}^{3}}\right)\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1.1472190365846532 \cdot 10^{-285}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\left(\frac{t}{b - y} + \left(\frac{y \cdot \frac{x}{z}}{b - y} + \frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}}\right)\right) - \left(\frac{t}{z} \cdot \frac{y}{{\left(b - y\right)}^{2}} + \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2.1850262770065685 \cdot 10^{+286}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))))