Average Error: 23.0 → 9.1
Time: 20.8s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -6.079573809220938 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\left(y + z \cdot b\right) - y \cdot z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2.465868948444032 \cdot 10^{-305}:\\ \;\;\;\;\frac{a}{y - b} + \left(\frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 3.881793583933535 \cdot 10^{+258}:\\ \;\;\;\;t \cdot \frac{z}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b} + \left(\frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)\\ \end{array}\]
\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:\\
\;\;\;\;x\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -6.079573809220938 \cdot 10^{-291}:\\
\;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\left(y + z \cdot b\right) - y \cdot z}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2.465868948444032 \cdot 10^{-305}:\\
\;\;\;\;\frac{a}{y - b} + \left(\frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 3.881793583933535 \cdot 10^{+258}:\\
\;\;\;\;t \cdot \frac{z}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - 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{a}{y - b} + \left(\frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)\\

\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))
   x
   (if (<=
        (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
        -6.079573809220938e-291)
     (/ (+ (* x y) (* z (- t a))) (- (+ y (* z b)) (* y z)))
     (if (<=
          (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
          2.465868948444032e-305)
       (+
        (/ 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))))
            3.881793583933535e+258)
         (+
          (* t (/ z (+ y (* z (- b y)))))
          (/ (- (* x y) (* z a)) (+ y (* z (- b y)))))
         (if (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) INFINITY)
           (/ x (- 1.0 z))
           (+
            (/ 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))))))))))))
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 = x;
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -6.079573809220938e-291) {
		tmp = ((x * y) + (z * (t - a))) / ((y + (z * b)) - (y * z));
	} else if ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 2.465868948444032e-305) {
		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)))) <= 3.881793583933535e+258) {
		tmp = (t * (z / (y + (z * (b - y))))) + (((x * y) - (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 = (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)))));
	}
	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

Original23.0
Target18.2
Herbie9.1
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

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

    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 0 35.4

      \[\leadsto \color{blue}{x}\]

    if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -6.0795738092209383e-291

    1. Initial program 0.4

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

    2. Taylor expanded around 0 0.4

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

    if -6.0795738092209383e-291 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.4658689484440322e-305 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

    1. Initial program 56.0

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

    3. Applied add-cube-cbrt_binary6456.0

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

    4. Applied associate-/l*_binary6456.0

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

    5. Simplified56.0

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

    6. Taylor expanded around 0 56.5

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

    7. Simplified56.0

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

    8. Taylor expanded around -inf 31.0

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

    9. Simplified14.2

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

    if 2.4658689484440322e-305 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 3.88179358393353459e258

    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 add-cube-cbrt_binary641.4

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

    4. Applied associate-/l*_binary641.4

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

    5. Simplified1.4

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

    6. Taylor expanded around 0 0.3

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

    7. Simplified0.3

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

    9. Applied *-un-lft-identity_binary640.3

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

    10. Applied times-frac_binary640.6

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

    if 3.88179358393353459e258 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

    1. Initial program 54.1

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

    2. Taylor expanded around inf 31.7

      \[\leadsto \color{blue}{\frac{x}{1 - z}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -6.079573809220938 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\left(y + z \cdot b\right) - y \cdot z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2.465868948444032 \cdot 10^{-305}:\\ \;\;\;\;\frac{a}{y - b} + \left(\frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 3.881793583933535 \cdot 10^{+258}:\\ \;\;\;\;t \cdot \frac{z}{y + z \cdot \left(b - y\right)} + \frac{x \cdot y - z \cdot a}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b} + \left(\frac{y}{z} \cdot \left(\frac{a}{{\left(y - b\right)}^{2}} - \frac{x}{y - b}\right) - \left(\frac{t}{y - b} + \frac{y \cdot t}{z \cdot {\left(y - b\right)}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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)))))