Average Error: 10.5 → 5.2
Time: 5.7s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z} \]
↓
\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{x}{t_1} - \frac{y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -1.20736653492792 \cdot 10^{-309}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_2 \leq 1.9822646599559802 \cdot 10^{+292}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array}\\ \end{array} \]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
t_1 := t - z \cdot a\\
t_2 := \frac{x - y \cdot z}{t_1}\\
\mathbf{if}\;t_2 \leq -\infty:\\
\;\;\;\;\frac{y}{a}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_3 := \frac{x}{t_1} - \frac{y \cdot z}{t_1}\\
\mathbf{if}\;t_2 \leq -1.20736653492792 \cdot 10^{-309}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;t_2 \leq 0:\\
\;\;\;\;\frac{y}{a}\\

\mathbf{elif}\;t_2 \leq 1.9822646599559802 \cdot 10^{+292}:\\
\;\;\;\;t_3\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{a}\\


\end{array}\\


\end{array}

(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ y a)
     (let* ((t_3 (- (/ x t_1) (/ (* y z) t_1))))
       (if (<= t_2 -1.20736653492792e-309)
         t_3
         (if (<= t_2 0.0)
           (/ y a)
           (if (<= t_2 1.9822646599559802e+292) t_3 (/ y 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 t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / a;
	} else {
		double t_3 = (x / t_1) - ((y * z) / t_1);
		double tmp_1;
		if (t_2 <= -1.20736653492792e-309) {
			tmp_1 = t_3;
		} else if (t_2 <= 0.0) {
			tmp_1 = y / a;
		} else if (t_2 <= 1.9822646599559802e+292) {
			tmp_1 = t_3;
		} else {
			tmp_1 = y / a;
		}
		tmp = tmp_1;
	}
	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

Original10.5
Target1.7
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -1.207366534927921e-309 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0 or 1.98226465995598025e292 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 41.4

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

    2. Taylor expanded in z around inf 20.1

      \[\leadsto \color{blue}{\frac{y}{a}} \]

    if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.207366534927921e-309 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.98226465995598025e292

    1. Initial program 0.2

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

    2. Applied div-sub_binary640.2

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

    3. Simplified0.2

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

    4. Simplified0.2

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.20736653492792 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 1.9822646599559802 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))