Average Error: 16.7 → 13.7
Time: 6.8s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.642073336958404 \cdot 10^{-86} \lor \neg \left(z \leq 1.3583837970033708 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{-\left(x + z \cdot \frac{y}{t}\right)}{-\left(\left(a + 1\right) + \frac{y}{t} \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z \cdot y}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.642073336958404 \cdot 10^{-86} \lor \neg \left(z \leq 1.3583837970033708 \cdot 10^{-139}\right):\\
\;\;\;\;\frac{-\left(x + z \cdot \frac{y}{t}\right)}{-\left(\left(a + 1\right) + \frac{y}{t} \cdot b\right)}\\

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

\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 (<= z -2.642073336958404e-86) (not (<= z 1.3583837970033708e-139)))
   (/ (- (+ x (* z (/ y t)))) (- (+ (+ a 1.0) (* (/ y t) b))))
   (* (+ x (/ (* z y) t)) (/ 1.0 (+ (+ a 1.0) (/ (* y b) t))))))
double code(double x, double y, double z, double t, double a, double b) {
	return (((double) (x + (((double) (y * z)) / t))) / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / t))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z <= -2.642073336958404e-86) || !(z <= 1.3583837970033708e-139))) {
		tmp = (((double) -(((double) (x + ((double) (z * (y / t))))))) / ((double) -(((double) (((double) (a + 1.0)) + ((double) ((y / t) * b)))))));
	} else {
		tmp = ((double) (((double) (x + (((double) (z * y)) / t))) * (1.0 / ((double) (((double) (a + 1.0)) + (((double) (y * b)) / 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

Original16.7
Target13.2
Herbie13.7
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.64207333695840391e-86 or 1.3583837970033708e-139 < z

    1. Initial program Error: 20.7 bits

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

    3. Applied add-cube-cbrtError: 21.0 bits

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

    4. Applied associate-/r*Error: 21.0 bits

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

    5. SimplifiedError: 18.9 bits

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

    7. Applied *-un-lft-identityError: 18.9 bits

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

    8. Applied associate-/r*Error: 18.9 bits

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

    9. SimplifiedError: 18.7 bits

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

    11. Applied frac-2negError: 18.7 bits

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

    12. SimplifiedError: 16.3 bits

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

    if -2.64207333695840391e-86 < z < 1.3583837970033708e-139

    1. Initial program Error: 8.3 bits

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

    3. Applied div-invError: 8.4 bits

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

  4. Final simplificationError: 13.7 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.642073336958404 \cdot 10^{-86} \lor \neg \left(z \leq 1.3583837970033708 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{-\left(x + z \cdot \frac{y}{t}\right)}{-\left(\left(a + 1\right) + \frac{y}{t} \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z \cdot y}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))