Average Error: 16.6 → 13.8
Time: 3.0s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.6642546363229162 \cdot 10^{-134} \lor \neg \left(z \le 1.2255216464227784 \cdot 10^{-277}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + \frac{y}{-t} \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.6642546363229162 \cdot 10^{-134} \lor \neg \left(z \le 1.2255216464227784 \cdot 10^{-277}\right):\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + \frac{y}{-t} \cdot \left(-b\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\end{array}
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 VAR;
	if (((z <= -1.6642546363229162e-134) || !(z <= 1.2255216464227784e-277))) {
		VAR = ((x + (z * (y / t))) / ((a + 1.0) + ((y / -t) * -b)));
	} else {
		VAR = ((x + (y / (t / z))) / ((a + 1.0) + (y / (t / b))));
	}
	return VAR;
}

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

Original16.6
Target13.1
Herbie13.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.6642546363229162e-134 or 1.2255216464227784e-277 < z

    1. Initial program 18.5

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

    3. Applied associate-/l*19.1

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

    5. Applied frac-2neg19.1

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

    6. Applied associate-/r/17.6

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

    8. Applied *-commutative17.6

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

    9. Applied associate-/l*15.3

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

    11. Applied div-inv15.4

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

    12. Simplified15.3

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

    if -1.6642546363229162e-134 < z < 1.2255216464227784e-277

    1. Initial program 7.1

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

    3. Applied associate-/l*6.5

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

    5. Applied associate-/l*6.5

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.6642546363229162 \cdot 10^{-134} \lor \neg \left(z \le 1.2255216464227784 \cdot 10^{-277}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + \frac{y}{-t} \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))