Average Error: 11.1 → 2.2
Time: 9.8s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq -6.0572448180137 \cdot 10^{-321} \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 2.0446532741568272 \cdot 10^{-287}\right) \land \frac{x - y \cdot z}{t - z \cdot a} \leq 3.645992002015573 \cdot 10^{+294}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq -6.0572448180137 \cdot 10^{-321} \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 2.0446532741568272 \cdot 10^{-287}\right) \land \frac{x - y \cdot z}{t - z \cdot a} \leq 3.645992002015573 \cdot 10^{+294}\right):\\
\;\;\;\;\frac{-y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (/ (- x (* y z)) (- t (* z a))) (- INFINITY))
         (not
          (or (<= (/ (- x (* y z)) (- t (* z a))) -6.0572448180137e-321)
              (and (not
                    (<=
                     (/ (- x (* y z)) (- t (* z a)))
                     2.0446532741568272e-287))
                   (<=
                    (/ (- x (* y z)) (- t (* z a)))
                    3.645992002015573e+294)))))
   (/ (- y) (- (/ t z) a))
   (- (/ x (- t (* z a))) (/ (* y z) (- t (* z a))))))
double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((x - (y * z)) / (t - (z * a))) <= -((double) INFINITY)) || !((((x - (y * z)) / (t - (z * a))) <= -6.0572448180137e-321) || (!(((x - (y * z)) / (t - (z * a))) <= 2.0446532741568272e-287) && (((x - (y * z)) / (t - (z * a))) <= 3.645992002015573e+294)))) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x / (t - (z * a))) - ((y * z) / (t - (z * a)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.1
Target1.8
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -6.05724e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.0446532741568272e-287 or 3.64599200201557323e294 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 40.7

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-inv_binary64_1678440.7

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

    4. Simplified40.7

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

    5. Taylor expanded around 0 42.5

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

    6. Simplified7.5

      \[\leadsto \color{blue}{\frac{-y}{\frac{t}{z} - a}}\]

    if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -6.05724e-321 or 2.0446532741568272e-287 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.64599200201557323e294

    1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z}\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{z \cdot y}{t - a \cdot z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq -6.0572448180137 \cdot 10^{-321} \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 2.0446532741568272 \cdot 10^{-287}\right) \land \frac{x - y \cdot z}{t - z \cdot a} \leq 3.645992002015573 \cdot 10^{+294}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))