Average Error: 16.5 → 9.5
Time: 10.6s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
↓
\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t_1 \leq -5.984824460587897 \cdot 10^{+215}:\\ \;\;\;\;\frac{x}{a + 1} - \left(\frac{y \cdot \left(x \cdot b\right)}{t \cdot {\left(a + 1\right)}^{2}} + \frac{y}{-1 - a} \cdot \frac{z}{t}\right)\\ \mathbf{elif}\;t_1 \leq 7.503591931827809 \cdot 10^{+298}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\\ \frac{x + \frac{z \cdot t_2}{\sqrt[3]{t}}}{\left(a + 1\right) + t_2 \cdot \frac{b}{\sqrt[3]{t}}} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\
\mathbf{if}\;t_1 \leq -5.984824460587897 \cdot 10^{+215}:\\
\;\;\;\;\frac{x}{a + 1} - \left(\frac{y \cdot \left(x \cdot b\right)}{t \cdot {\left(a + 1\right)}^{2}} + \frac{y}{-1 - a} \cdot \frac{z}{t}\right)\\

\mathbf{elif}\;t_1 \leq 7.503591931827809 \cdot 10^{+298}:\\
\;\;\;\;\begin{array}{l}
t_2 := \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\\
\frac{x + \frac{z \cdot t_2}{\sqrt[3]{t}}}{\left(a + 1\right) + t_2 \cdot \frac{b}{\sqrt[3]{t}}}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -5.984824460587897e+215)
     (-
      (/ x (+ a 1.0))
      (+
       (/ (* y (* x b)) (* t (pow (+ a 1.0) 2.0)))
       (* (/ y (- -1.0 a)) (/ z t))))
     (if (<= t_1 7.503591931827809e+298)
       (let* ((t_2 (/ y (* (cbrt t) (cbrt t)))))
         (/ (+ x (/ (* z t_2) (cbrt t))) (+ (+ a 1.0) (* t_2 (/ b (cbrt t))))))
       (/ z b)))))
double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -5.984824460587897e+215) {
		tmp = (x / (a + 1.0)) - (((y * (x * b)) / (t * pow((a + 1.0), 2.0))) + ((y / (-1.0 - a)) * (z / t)));
	} else if (t_1 <= 7.503591931827809e+298) {
		double t_2 = y / (cbrt(t) * cbrt(t));
		tmp = (x + ((z * t_2) / cbrt(t))) / ((a + 1.0) + (t_2 * (b / cbrt(t))));
	} else {
		tmp = z / b;
	}
	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

Original16.5
Target13.1
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.98482446058789688e215

    1. Initial program 39.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary6439.6

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

    4. Applied associate-/r*_binary6439.6

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

    5. Simplified28.2

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

    6. Taylor expanded around -inf 43.0

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

    7. Simplified33.9

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

    if -5.98482446058789688e215 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 7.5035919318278092e298

    1. Initial program 6.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary646.8

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

    4. Applied associate-/r*_binary646.8

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

    5. Simplified8.2

      \[\leadsto \frac{x + \frac{\color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot z}}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    6. Using strategy rm

    7. Applied add-cube-cbrt_binary648.2

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

    8. Applied times-frac_binary646.8

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

    if 7.5035919318278092e298 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

    1. Initial program 63.1

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

    2. Taylor expanded around inf 13.0

      \[\leadsto \color{blue}{\frac{z}{b}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5.984824460587897 \cdot 10^{+215}:\\ \;\;\;\;\frac{x}{a + 1} - \left(\frac{y \cdot \left(x \cdot b\right)}{t \cdot {\left(a + 1\right)}^{2}} + \frac{y}{-1 - a} \cdot \frac{z}{t}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 7.503591931827809 \cdot 10^{+298}:\\ \;\;\;\;\frac{x + \frac{z \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))