Average Error: 10.8 → 1.8
Time: 8.6s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z} \]
↓
\[\begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;z \leq -5.011713591805735 \cdot 10^{-75} \lor \neg \left(z \leq 6.890244941997684 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{x}{t_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t_1}\\ \end{array} \]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
t_1 := t - z \cdot a\\
\mathbf{if}\;z \leq -5.011713591805735 \cdot 10^{-75} \lor \neg \left(z \leq 6.890244941997684 \cdot 10^{-165}\right):\\
\;\;\;\;\frac{x}{t_1} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{x - z \cdot y}{t_1}\\


\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))))
   (if (or (<= z -5.011713591805735e-75) (not (<= z 6.890244941997684e-165)))
     (- (/ x t_1) (/ y (- (/ t z) a)))
     (/ (- x (* z y)) t_1))))
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 tmp;
	if ((z <= -5.011713591805735e-75) || !(z <= 6.890244941997684e-165)) {
		tmp = (x / t_1) - (y / ((t / z) - a));
	} else {
		tmp = (x - (z * y)) / t_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

Original10.8
Target1.6
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.01171359180573526e-75 or 6.890244941997684e-165 < z

    1. Initial program 15.4

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

    2. Taylor expanded in x around 0 15.4

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

    3. Applied associate-/l*_binary649.8

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

    4. Taylor expanded in t around 0 2.5

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

    if -5.01171359180573526e-75 < z < 6.890244941997684e-165

    1. Initial program 0.1

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

    2. Applied pow1_binary640.1

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.011713591805735 \cdot 10^{-75} \lor \neg \left(z \leq 6.890244941997684 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))