Average Error: 23.3 → 8.3
Time: 25.3s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\left(\frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{t}{{\left(b - y\right)}^{2}} + \frac{x}{y - b}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -8.868229609988521 \cdot 10^{-308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\left(\frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{t}{{\left(b - y\right)}^{2}} + \frac{x}{y - b}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 9.918042372221392 \cdot 10^{+290}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array}\]
\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(\frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{t}{{\left(b - y\right)}^{2}} + \frac{x}{y - b}\right)\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -8.868229609988521 \cdot 10^{-308}:\\
\;\;\;\;\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{y \cdot a}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{t}{{\left(b - y\right)}^{2}} + \frac{x}{y - b}\right)\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 9.918042372221392 \cdot 10^{+290}:\\
\;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - 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))
   (-
    (+ (/ (* y a) (* z (pow (- b y) 2.0))) (/ a (- y b)))
    (+ (/ t (- y b)) (* (/ y z) (+ (/ t (pow (- b y) 2.0)) (/ x (- y b))))))
   (if (<=
        (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
        -8.868229609988521e-308)
     (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
     (if (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) 0.0)
       (-
        (+ (/ (* y a) (* z (pow (- b y) 2.0))) (/ a (- y b)))
        (+
         (/ t (- y b))
         (* (/ y z) (+ (/ t (pow (- b y) 2.0)) (/ x (- y b))))))
       (if (<=
            (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
            9.918042372221392e+290)
         (+
          (/ (* z t) (+ y (* z (- b y))))
          (/ (- (* x y) (* 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 = (((y * a) / (z * pow((b - y), 2.0))) + (a / (y - b))) - ((t / (y - b)) + ((y / z) * ((t / pow((b - y), 2.0)) + (x / (y - b)))));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -8.868229609988521e-308) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 0.0) {
		tmp = (((y * a) / (z * pow((b - y), 2.0))) + (a / (y - b))) - ((t / (y - b)) + ((y / z) * ((t / pow((b - y), 2.0)) + (x / (y - b)))));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 9.918042372221392e+290) {
		tmp = ((z * t) / (y + (z * (b - y)))) + (((x * y) - (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

Original23.3
Target17.9
Herbie8.3
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

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 or -8.86822960998852061e-308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

    1. Initial program 55.1

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

    3. Applied add-cube-cbrt_binary64_2057355.1

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

    4. Taylor expanded around -inf 28.4

      \[\leadsto \color{blue}{\left(\frac{a \cdot y}{{\left({\left(\sqrt[3]{-1}\right)}^{3} \cdot b - {\left(\sqrt[3]{-1}\right)}^{3} \cdot y\right)}^{2} \cdot z} + \frac{a}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot b - {\left(\sqrt[3]{-1}\right)}^{3} \cdot y}\right) - \left(\frac{t}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot b - {\left(\sqrt[3]{-1}\right)}^{3} \cdot y} + \left(\frac{t \cdot y}{{\left({\left(\sqrt[3]{-1}\right)}^{3} \cdot b - {\left(\sqrt[3]{-1}\right)}^{3} \cdot y\right)}^{2} \cdot z} + \frac{x \cdot y}{\left({\left(\sqrt[3]{-1}\right)}^{3} \cdot b - {\left(\sqrt[3]{-1}\right)}^{3} \cdot y\right) \cdot z}\right)\right)}\]

    5. Simplified20.1

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

    if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -8.86822960998852061e-308

    1. Initial program 0.4

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

    if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9180423722213923e290

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 9.9180423722213923e290 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

    1. Initial program 62.5

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

    2. Taylor expanded around inf 21.5

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\left(\frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{t}{{\left(b - y\right)}^{2}} + \frac{x}{y - b}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -8.868229609988521 \cdot 10^{-308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\left(\frac{y \cdot a}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{t}{{\left(b - y\right)}^{2}} + \frac{x}{y - b}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 9.918042372221392 \cdot 10^{+290}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))