Average Error: 16.8 → 13.0
Time: 5.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.67772489405608202 \cdot 10^{32} \lor \neg \left(y \le 1.153339796061393 \cdot 10^{76} \lor \neg \left(y \le 1.2925577614671244 \cdot 10^{272}\right)\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{y}{t} \cdot b}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.67772489405608202 \cdot 10^{32} \lor \neg \left(y \le 1.153339796061393 \cdot 10^{76} \lor \neg \left(y \le 1.2925577614671244 \cdot 10^{272}\right)\right):\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{y}{t} \cdot 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 temp;
	if (((y <= -6.677724894056082e+32) || !((y <= 1.153339796061393e+76) || !(y <= 1.2925577614671244e+272)))) {
		temp = ((x + (y * (z / t))) / ((a + 1.0) + (y / (t / b))));
	} else {
		temp = (fma((y / t), z, x) / ((a + 1.0) + ((y / t) * b)));
	}
	return temp;
}

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

Original16.8
Target13.6
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -6.677724894056082e+32 or 1.153339796061393e+76 < y < 1.2925577614671244e+272

    1. Initial program 32.1

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

    3. Applied associate-/l*28.7

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

    5. Applied *-un-lft-identity28.7

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

    6. Applied times-frac23.0

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

    7. Simplified23.0

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

    if -6.677724894056082e+32 < y < 1.153339796061393e+76 or 1.2925577614671244e+272 < y

    1. Initial program 7.5

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

    3. Applied associate-/l*10.1

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

    5. Applied associate-/r/7.3

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

    7. Applied *-un-lft-identity7.3

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

    8. Applied associate-/r*7.3

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

    9. Simplified6.9

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.67772489405608202 \cdot 10^{32} \lor \neg \left(y \le 1.153339796061393 \cdot 10^{76} \lor \neg \left(y \le 1.2925577614671244 \cdot 10^{272}\right)\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{y}{t} \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))