Average Error: 16.4 → 2.7
Time: 9.7s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.7954477689789754 \cdot 10^{-82} \lor \neg \left(y \leq 1.677139076070172 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{z}{b + \frac{t}{y} \cdot \left(a + 1\right)} + \frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a + \left(1 + \frac{y \cdot b}{t}\right)\right)} + \frac{x}{a + \left(1 + \left(y \cdot b\right) \cdot \frac{1}{t}\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.7954477689789754 \cdot 10^{-82} \lor \neg \left(y \leq 1.677139076070172 \cdot 10^{-49}\right):\\
\;\;\;\;\frac{z}{b + \frac{t}{y} \cdot \left(a + 1\right)} + \frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\

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

\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
 (if (or (<= y -1.7954477689789754e-82) (not (<= y 1.677139076070172e-49)))
   (+ (/ z (+ b (* (/ t y) (+ a 1.0)))) (/ x (+ a (+ 1.0 (* y (/ b t))))))
   (+
    (/ (* y z) (* t (+ a (+ 1.0 (/ (* y b) t)))))
    (/ x (+ a (+ 1.0 (* (* y b) (/ 1.0 t))))))))
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 tmp;
	if ((y <= -1.7954477689789754e-82) || !(y <= 1.677139076070172e-49)) {
		tmp = (z / (b + ((t / y) * (a + 1.0)))) + (x / (a + (1.0 + (y * (b / t)))));
	} else {
		tmp = ((y * z) / (t * (a + (1.0 + ((y * b) / t))))) + (x / (a + (1.0 + ((y * b) * (1.0 / t)))));
	}
	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

Original16.4
Target13.3
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.7954477689789754e-82 or 1.6771390760701721e-49 < y

    1. Initial program 25.6

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

    2. Taylor expanded around 0 23.2

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

    3. Simplified23.2

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

    5. Applied associate-/l*_binary6419.3

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

    6. Simplified18.2

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

    7. Taylor expanded around 0 6.8

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

    8. Simplified5.1

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

    10. Applied *-un-lft-identity_binary645.1

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

    11. Applied times-frac_binary643.2

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

    12. Simplified3.2

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

    if -1.7954477689789754e-82 < y < 1.6771390760701721e-49

    1. Initial program 2.5

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

    2. Taylor expanded around 0 2.1

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

    3. Simplified2.1

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

    5. Applied div-inv_binary642.1

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7954477689789754 \cdot 10^{-82} \lor \neg \left(y \leq 1.677139076070172 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{z}{b + \frac{t}{y} \cdot \left(a + 1\right)} + \frac{x}{a + \left(1 + y \cdot \frac{b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t \cdot \left(a + \left(1 + \frac{y \cdot b}{t}\right)\right)} + \frac{x}{a + \left(1 + \left(y \cdot b\right) \cdot \frac{1}{t}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021168 
(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))))