Average Error: 16.0 → 15.1
Time: 9.6s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -2.0427366300603076 \cdot 10^{+57}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a - 1\right) \cdot \left(y + \frac{t}{b} \cdot \left(a + 1\right)\right)} \cdot \left(\left(a - 1\right) \cdot \frac{t}{b}\right)\\ \mathbf{elif}\;y \leq 255.45322304609837:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y}{t}}{\frac{1}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -2.0427366300603076 \cdot 10^{+57}:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a - 1\right) \cdot \left(y + \frac{t}{b} \cdot \left(a + 1\right)\right)} \cdot \left(\left(a - 1\right) \cdot \frac{t}{b}\right)\\

\mathbf{elif}\;y \leq 255.45322304609837:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y}{t}}{\frac{1}{b}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{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
 (if (<= y -2.0427366300603076e+57)
   (*
    (/ (+ x (* y (/ z t))) (* (- a 1.0) (+ y (* (/ t b) (+ a 1.0)))))
    (* (- a 1.0) (/ t b)))
   (if (<= y 255.45322304609837)
     (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (/ y t) (/ 1.0 b))))
     (/ (+ x (/ y (/ t z))) (+ (+ a 1.0) (/ y (/ t 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 tmp;
	if (y <= -2.0427366300603076e+57) {
		tmp = ((x + (y * (z / t))) / ((a - 1.0) * (y + ((t / b) * (a + 1.0))))) * ((a - 1.0) * (t / b));
	} else if (y <= 255.45322304609837) {
		tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y / t) / (1.0 / b)));
	} else {
		tmp = (x + (y / (t / z))) / ((a + 1.0) + (y / (t / 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

Original16.0
Target12.8
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.04273663006030763e57

    1. Initial program 33.7

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

    3. Applied associate-/l*_binary64_1210129.7

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

    5. Applied *-un-lft-identity_binary64_1203729.7

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

    6. Applied times-frac_binary64_1203224.2

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

    7. Simplified24.2

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

    9. Applied flip-+_binary64_1206232.4

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

    10. Applied frac-add_binary64_1202742.0

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

    11. Applied associate-/r/_binary64_1210239.2

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

    12. Simplified37.9

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

    if -2.04273663006030763e57 < y < 255.453223046098373

    1. Initial program 4.2

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

    3. Applied associate-/l*_binary64_121017.3

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

    5. Applied div-inv_binary64_120387.3

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

    6. Applied associate-/r*_binary64_121004.1

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

    if 255.453223046098373 < y

    1. Initial program 29.3

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

    3. Applied associate-/l*_binary64_1210126.5

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

    5. Applied associate-/l*_binary64_1210122.2

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.0427366300603076 \cdot 10^{+57}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a - 1\right) \cdot \left(y + \frac{t}{b} \cdot \left(a + 1\right)\right)} \cdot \left(\left(a - 1\right) \cdot \frac{t}{b}\right)\\ \mathbf{elif}\;y \leq 255.45322304609837:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y}{t}}{\frac{1}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(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))))